Teaching Mathematical Reasoning in Secondary School Classrooms ( PDFDrive.com )

Teaching Mathematical Reasoning in Secondary School Classrooms

Karin Brodie Teaching Mathematical Reasoning in Secondary School Classrooms With Contributions by Kurt Coetzee Lorraine Lauf Stephen Modau Nico Molefe Romulus O’Brien iii

Karin Brodie School of Education University of the Witwatersrand Johannesburg South Africa [email protected] ISBN 978-0-387-09741-1 e-ISBN 978-0-387-09742-8 DOI 10.1007/978-0-387-09742-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009935695 © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Foreword The Road to Reasoning The teachers in this book share a worthy and courageous mission. They have all set out to provide children with one of the most important educational experiences it is possible to have – a form of mathematics teaching that is based upon sense making and discussion, rather than submission and silence. Mathematical “reasoning” is what mathematicians do – it involves forming and communicating a path between one idea or concept and the next. When students form these paths they come to enjoy mathe- matics, understand the reasons why ideas work, and develop a connected and power- ful form of knowledge. When students do not engage in reasoning, they often do not know that there are paths between different ideas in mathematics and they come to believe, dangerously, that mathematics is a set of isolated facts and methods that need to be remembered. I have visited hundreds of classrooms across the world in which students have been required to work in silence on maths questions, never talking about the ideas or forming links and connections between ideas; most of these stu- dents come to dislike mathematics and drop the subject as soon as they can. Such students are not only being denied the opportunity to learn in the most helpful way, but they are denied access to real, living mathematics. The teachers in this book, through their work with Karin Brodie, the author, learned about the value of mathematical reasoning and set out to teach students to engage in this valuable act. This book shares their important journey and provides the world with new lenses for considering the teaching acts that were involved, as well as the challenges and obstacles that stood in their way. For whilst we know the importance of reasoning to children’s mathematical futures it would be dishonest to pretend that teaching approaches that invite students to communicate their mathe- matical thoughts and make connections between ideas are easy or well understood. We have reached an advanced stage in the development of education and yet, incredibly, we are still relatively uninformed about the ways teachers of mathemat- ics can teach students to reason, which is part of the reason this book is so valuable and could be a wonderful resource for many. When Deborah Ball, in the United States, then an elementary teacher of math- ematics, now a university dean, released a videotape of her teaching 7- and 8-year olds to reason about odd and even numbers, the world was shocked to witness a boy v

vi Foreword named Shea propose a new way of classifying odd numbers. His numbers – those that can be grouped into even numbers of pairs of twos – came to be known as “Shea numbers”. The rich conversations in which the young children engaged in the mathematics class that appeared on tape, seemed to unfold effortlessly, although in reality they were expertly choreographed by the teacher. Deborah Ball has offered records of her teaching decisions and actions, which have been read by scores of people worldwide, including the teachers who write in this book. She was one of the first teachers to offer such valuable records and analyses. This book adds to the small but important collection of teachers who have engaged students in mathematical reasoning and documented and unpacked the important teaching acts that took place. But what makes a record of teaching useful and worthwhile? Every act of teach- ing, with a classroom full of children and their many thoughts and actions, is extremely complex, and descriptions of a class in action can remain highly contex- tualized and difficult for others to learn from. A teacher may record thoughts and moves without communicating them in such a way that they are useful for other teachers, educators, and analysts. The art in producing a record that is powerful and valuable for others comes partly from having important teaching experiences to talk about and partly from having a way of raising the individual acts to a higher and more generalizable level that other teachers can learn from. This is where the com- bination of the reports of the teachers who engaged students in reasoning, and the theoretical lenses applied by Karin, are so generative and fruitful for the rest of the world to learn from. When a new idea and teaching act is connected with a theory of learning, the result can be very powerful indeed. An example of the way a teaching act can be named and made more general is the case of a set of interactions that has become known as IRE. These describe a common teaching situation when a teacher initiates something (I), elicits a response from a student (R), and then evaluates the response (E). Researchers found that the majority of the interactions that take place in classrooms follow the IRE response pattern and they gave it a particular classification. Since that initial classification IRE has been used by scores of researchers and analysts over many years and has proved extremely useful in the advancement of teaching. Yet teaching classifica- tions such as IRE are rare and the field of mathematics education has not benefitted from a similar mapping and classification of the teaching interactions that take place when students are taught to reason about mathematics. This book provides such a mapping. Karin notes that a reasoning approach to mathematics involves a change in authority. Students no longer need to look to teachers or textbooks to know if they are moving in the right directions in mathematics, as they have learned a set of reasons and connections that they can refer back to, evaluating their own thoughts and ideas. This may seem as though the authority is shifting from the teacher to the students and this is partly true, but it is important to note that the authority is also shifting from the teacher to the domain of mathematics itself. Students no longer need to refer to teachers to evaluate their mathematical thoughts, because they can refer to the domain of mathematics, to consider whether they have followed the

Foreword vii correct connections and paths. This is just one way in which reasoning as an act brings classrooms closer to real and living mathematics. In addition, we now have evidence that when students receive opportunities to discuss mathematics and express their own thoughts, they become more open-minded as they learn to be appreciative and respectful of other people’s ideas. Mathematical reasoning encour- ages respect, responsibility, and a personal empowerment that has long been miss- ing in mathematics classrooms. Karin starts this book by quoting the goals of the new South African curriculum – to heal the divisions of the past and build a human rights culture. Mathematics, the subject so many believe to be abstract and removed from such responsibilities, has a key role to play in promoting such a culture, in South Africa and beyond. This book communicates the way that mathematics can provide this valuable contribution and the important work of teachers in doing so. I hope you enjoy it and use it as both inspiration and resource. Jo Boaler The University of Sussex

Contents Introduction to Part 1...................................................................................... 1 1 Teaching Mathematical Reasoning: A Challenging Task....................... 7 The Centrality of Mathematical Reasoning in Mathematics Education...... 7 Justifying and Generalizing..................................................................... 8 The Role of Proof in Mathematical Reasoning....................................... 9 Creativity and Reasoning........................................................................ 10 12 Theories of Learning and Mathematical Reasoning.................................... 12 Constructivism......................................................................................... 14 Socio-Cultural Theories........................................................................... 16 Situated Theories..................................................................................... 18 19 Teaching Mathematical Reasoning.............................................................. 20 Tasks for Mathematical Reasoning.......................................................... 22 Classroom Interaction.............................................................................. The Challenges of Teaching Mathematical Reasoning............................... 2 Contexts, Resources, and Reform............................................................ 23 Responses to Reforms................................................................................. 23 The South African Context.......................................................................... 26 Five Schools: Contexts and Resources........................................................ 28 28 Race and Socio-Economic Status............................................................ 29 School Resources..................................................................................... 31 Classroom Resources............................................................................... 33 Learner Knowledge..................................................................................... 35 The Tasks..................................................................................................... 35 The Grade 11 Tasks................................................................................. 36 The Grade 10 Tasks................................................................................. ix

x Contents Introduction to Part 2...................................................................................... 39 3 Mathematical Reasoning Through Tasks: Learners’ Responses........... 43 Tasks that Support Mathematical Reasoning.............................................. 44 Teaching for Mathematical Reasoning........................................................ 46 The Classroom and the Tasks...................................................................... 47 Learners’ Responses: An Overview............................................................. 48 Learners’ Responses: Detailed Analysis...................................................... 49 Teacher–Learner Interactions...................................................................... 52 52 Encouraging Participation....................................................................... 53 Using the Contribution to Move Forward............................................... 54 Pushing for Explanation of Particular Ideas............................................ 55 Conclusions and Implications...................................................................... 4 Learning Mathematical Reasoning in a Collaborative 57 Whole-Class Discussion............................................................................. 58 What Is Mathematical Reasoning?.............................................................. 59 Why Teach Mathematical Reasoning?........................................................ 60 Collaborative Learning and Mathematical Reasoning................................ 61 Summarizing My Perspective...................................................................... 62 My Classroom............................................................................................. 62 The Analysis................................................................................................ 63 Winile’s Learning........................................................................................ 64 64 Making Observations............................................................................... 65 Explaining and Justifying Assertions Made............................................ 67 Connecting Observations with Mathematical Representations............... 68 Reconstructing Conceptual Understanding............................................. 69 Testing Other Claims............................................................................... 69 The Teacher’s Role...................................................................................... 70 Establishing Discourse............................................................................ 70 Framing Discussion................................................................................. 71 Lesson Flow or Momentum..................................................................... Conclusions and Implications...................................................................... 5 Classroom Practices for Teaching and Learning 73 Mathematical Reasoning........................................................................... 74 Classroom Practices..................................................................................... 75 Learning Mathematical Reasoning.............................................................. 76 Teaching Mathematical Reasoning: Questioning and Listening................. 78 My Classroom............................................................................................. 79 Teacher Moves and Practices...................................................................... 82 Learner Moves and Practices....................................................................... 84 Conclusions and Implications......................................................................

Contents xi 6 Teaching Mathematical Reasoning with the Five Strands..................... 87 A Social-Constructivist Framework............................................................ 88 Mathematical Practices and Proficiency...................................................... 89 My Classroom and the Tasks....................................................................... 90 Initial Analysis............................................................................................. 94 Classroom Interaction.............................................................................. 94 Learners’ Work........................................................................................ 95 The Five Strands in the Lesson................................................................... 96 Procedural Fluency.................................................................................. 96 Conceptual Understanding...................................................................... 97 Strategic Competence.............................................................................. 98 Adaptive Reasoning................................................................................. 99 The Five Strands in the Learners’ Work...................................................... 99 Conclusion................................................................................................... 100 7 Teaching the Practices of Justification and Explanation....................... 103 Construction and Practices.......................................................................... 104 The Practices of Justification and Explanation............................................ 104 The Importance of Tasks............................................................................. 106 The Teacher’s Contribution......................................................................... 106 My Classroom............................................................................................. 108 The Learners’ Written Responses................................................................ 109 Whole-Class Interaction.............................................................................. 111 Incorrect Justification.............................................................................. 112 Partial Justification.................................................................................. 114 Correct Justification................................................................................. 115 Conclusions................................................................................................. 117 Introduction to Part 3...................................................................................... 119 8 Learner Contributions.............................................................................. 121 Learner Contributions and Mathematical Reasoning.................................. 122 Describing Learner Contributions............................................................... 123 Distribution of Learner Contributions......................................................... 124 Accounting for Learner Contributions........................................................ 126 Basic Errors............................................................................................. 127 Appropriate Errors................................................................................... 128 Missing Information................................................................................ 130 Partial Insights......................................................................................... 131 Complete, Correct Contributions............................................................. 132 Going Beyond the Task........................................................................... 134 Summary...................................................................................................... 136

xii Contents   9 Teacher Responses to Learner Contributions....................................... 139 Teacher Moves........................................................................................... 139 Distributions of Teacher Moves................................................................ 142 Mainly Maintaining: Mr. Nkomo.............................................................. 142 The Power of Inserting: Ms. King............................................................. 145 Strategic Combinations: Mr. Daniels........................................................ 149 Supporting Learner Moves: Mr. Mogale................................................... 153 Entertaining Errors: Mr. Peters.................................................................. 157 Overview: Teacher Responses to Learner Contributions.......................... 160 Trajectories for Working with Learners’ Contributions............................. 163 10 Dilemmas of Teaching Mathematical Reasoning.................................. 167 Teaching Dilemmas................................................................................... 167 Linking Learners with the Subject........................................................ 168 Working Simultaneously with Individuals and Groups......................... 169 The “Press” Move...................................................................................... 170 To Press or Not to Press?........................................................................... 172 To Take Up or Ignore Learners’ Contributions?........................................ 176 Conclusions............................................................................................... 179 11 Learner Resistance to Teacher Change................................................. 183 Resistance to Pedagogy............................................................................. 183 The Context of the Resistance................................................................... 187 Learner Resistance..................................................................................... 191 The Teacher’s Contributions...................................................................... 193 Making Sense of the Resistance................................................................ 196 12 Conclusions and Ways Forward: The “Messy” Middle Ground........ 199 Tasks and Mathematical Reasoning.......................................................... 200 Supporting Learner Contributions............................................................. 201 Working with Learner Errors..................................................................... 202 Classroom Conversations.......................................................................... 202 Maintaining the IRE/F............................................................................... 203 Supporting all Learners to Participate....................................................... 204 Learner Resistance..................................................................................... 205 Conclusions............................................................................................... 205 Appendix.......................................................................................................... 207 References......................................................................................................... 213 Index................................................................................................................. 223

List of Tables Table 2.1  Demographics of schools............................................................. 29 Table 2.2  Resources available at the schools.............................................. 30 Table 2.3  Description of research classes.................................................... 31 Table 2.4  Variation across schools............................................................... 32 Table 2.5  Variation across schools............................................................... 34 Table 2.6 Variation across teachers in tasks, learner 38 knowledge and SES...................................................................... Table 3.1  Correct and incorrect responses................................................. 48 Table 3.2  Groups responses to question 3................................................... 50 Table 5.1  Teacher moves............................................................................... 79 Table 5.2  Learner moves.............................................................................. 82 Table 6.1  Strands in classroom activities.................................................... 94 Table 6.2  Evidence of strands in learners’ work........................................ 96 Table 7.1  Justifications for the conjecture being true................................ 110 Table 8.1  Examples of different kinds of contributions............................ 123 Table 8.2  Distributions of learner contributions across the classrooms............................................................................... 125 Table 8.3  Variation across teachers in tasks, learner knowledge and SES...................................................................... 126 Table 8.4  Key variables and learner contributions.................................... 136 Table 9.1  Subcategories of “follow up”....................................................... 140 Table 9.2  Subcategories of “follow up”....................................................... 141 Table 9.3  Teacher moves and learner contributions (part 1).................... 161 Table 9.4  Teacher moves and learner contributions (part 2).................... 161 xiii

Chapter 1 Teaching Mathematical Reasoning: A Challenging Task The Centrality of Mathematical Reasoning in Mathematics Education When we “reason”, we develop lines of thinking or argument, which might serve a number of purposes – to convince others or ourselves of a particular claim; to solve a problem; or to integrate a number of ideas into a more coherent whole. Two pro- cesses are important to reasoning – first, that the different steps or moves in the line of reasoning are connected with each other (not necessarily analytically or deduc- tively); and second, that these links are somehow “reasoned”, there are reasons why one move follows another and how a number of moves come together to form an argument or to solve a problem (Ball and Bass 2003). Brousseau and Gibel (2005) point out that these reasons are only considered to be reasonable when they relate to the constraints of the problem or the knowledge under consideration. An appeal to authority, for example to what a teacher or textbook says, does not count as a reason for a productive argument. The product of a reasoning process is a text, either spoken or written (Douek 2005), which presents warrants for a conclusion that is acceptable within the com- munity that is producing the argument (Krummheuer 1995). An individual can reason, or a group of people can reason together, co-producing the line of argu- ment1. Mathematical reasoning assumes mathematical communication (Ball and Bass 2003; Douek 2005; Krummheuer 1995). Communication is an integral part of the process of reasoning, both for an individual working with previously produced texts to produce a new one, and for groups working together to produce an argu- ment. The texts or products of reasoning have, as their main purpose, to communi- cate reasoning. Mathematical reasoning is reasoning about and with the objects of mathematics. However, the relationship between mathematical reasoning and mathematics is not obvious (Steen 1999), and the processes involved in mathematical reasoning need 1 Social perspectives on learning and thinking would argue that even an individual reasoning, seemingly on her own, is in fact in dialogue with others, co-producing an argument, with an imagined audience, with ideas from others, and in a social and historical context (see below). K. Brodie, Teaching Mathematical Reasoning in Secondary School Classrooms, 7 DOI 10.1007/978-0-387-09742-8_1, © Springer Science+Business Media, LLC 2010

8 1  Teaching Mathematical Reasoning: A Challenging Task some elaboration. For Ball and Bass (2003) reasoning is a “basic skill” (p. 28) of mathematics and is necessary for a number of purposes – to understand mathemati- cal concepts, to use mathematical ideas and procedures flexibly, and to reconstruct once understood, but forgotten mathematical knowledge. Kilpatrick et  al. (2001) define a notion of mathematical proficiency which requires five intertwined and mutually influential strands – conceptual understanding, which entails comprehen- sion of mathematical concepts, operations, and relations; procedural fluency, involving skill in carrying out procedures flexibly, accurately, efficiently, and appropriately; strategic competence, which is the ability to formulate, represent, and solve mathematical problems; adaptive reasoning, which is the capacity for logical thought, reflection, explanation, and justification; and productive disposi- tion, an orientation to seeing mathematics as sensible, useful, worthwhile, and reasonable, and that anyone can reason to make sense of mathematical ideas2. For Kilpatrick et al. (2001), although all the strands are important and mutually influ- ential, “adaptive reasoning is the glue that holds everything together” (p. 129) in that it allows for concepts and procedures to connect together in sensible ways, suggests possibilities for problem solving, and allows for disagreements to be set- tled in reasoned ways. Central to adaptive reasoning is the justification of claims and development of arguments. This view of mathematical proficiency has informed all of the work in this book. Most directly, in Chap. 6 we reflect on one teacher’s attempt to teach the five strands in a holistic way. The teacher found that she devoted most of the time to conceptual understanding rather than procedural fluency, which is traditionally the norm in mathematics classrooms (Kilpatrick et al. 2001; Schoenfeld 1988; Stigler and Hiebert 1999). However, she was concerned that she devoted less time to stra- tegic competence and adaptive reasoning. She also found that more than half of the learners in her class showed evidence of all five strands in their written work. In Chaps. 5 and 7 we focus on the strand of adaptive reasoning and show how two teachers supported learners to reason adaptively. Justifying and Generalizing The literature suggests that there are two key practices involved in mathematical reasoning – justifying and generalizing – and other mathematical practices such as symbolizing, representing, and communicating, are key in supporting these (Ball 2003; Ball and Bass 2003; Davis and Maher 1997; Triandafillidis and Potari 2005). For Kilpatrick et al. justifying is a key element of adaptive reasoning and to justify means “to provide sufficient reason for” (p. 130). They argue “students need to be able to justify and explain ideas in order to make their reasoning clear, hone their 2 I note here that Kilpatrick et al.’s work is an extension of the more usual distinctions of concep- tual and procedural understandings of mathematics (Hiebert and Lefevre, 1986).

The Centrality of Mathematical Reasoning in Mathematics Education 9 reasoning skills and improve their conceptual understanding” (p. 130). For Ball and Bass, “unjustified knowledge is unreasoned and, hence, easily becomes unreason- able” (p. 29). Justification is a key mathematical practice that allows mathemati- cians and mathematics teachers and learners to make connections between different ideas and parts of an argument, to provide warrant for claims and conjectures, to settle disputes, and to develop new mathematical ideas. For Russell (1999), mathematical reasoning is “essentially about the develop- ment, justification and use of mathematical generalizations” (p. 1). These general- izations create an interconnected web of mathematical knowledge – conceptual understanding in Kilpatrick et  al.’s terms. For Russell, “seeing mathematics as a web of interrelated ideas is both a result of an emphasis on mathematical reasoning and a foundation for reasoning further” (p. 5). Creating generalizations also enables problem solving, as generalizations support learners to see the underlying structure of the problem and the bigger class of problems or ideas that it instantiates (Brousseau and Gibel 2005; Kilpatrick et  al. 2001; Russell 1999). Russell also introduces a notion of “mathematical memory”, which is a memory of fundamental mathematical relationships, rather than of isolated facts. This kind of memory is what allows mathematical knowers to reconstruct, in a reasoned way, mathematical concepts, procedures, and principles that they might have forgotten (Ball and Bass 2003; Brousseau and Gibel 2005). It also supports sense making and insight in mathematics, and creates the conditions for solving problems. In Chap. 7, we directly address the challenges that a teacher faced in supporting his learners to justify their thinking. The vast majority of learners in his class were not able to answer the question: “can x2+1 be less than zero, when x is a real num- ber”, with appropriate justifications. We show how the teacher worked through a number of different contributions from learners, ranging from incorrect justifica- tions through those that were partially correct, to one that was completely correct, asking them to discuss and communicate their reasoning. Even though his learners had very weak mathematical knowledge, they were, with a lot of help from the teacher, able to contribute and to help each other develop better justifications. In each of the other teachers’ chapters, we see examples of learners’ successes and challenges as they work to justify, explain, and generalize their ideas. The Role of Proof in Mathematical Reasoning Justification and generalization are closely related to proof in mathematics. In fact, for many mathematicians and in many mathematics curricula, mathematical rea- soning is equated with proof. In this book we take the view, together with others (Ball and Bass 2003; Davis and Hersh 1981; Hanna and Jahnke 1996; Kilpatrick et  al. 2001; Kline 1980; Krummheuer 1995), that whereas proof is one form of argument and justification, not all arguments and justifications are proofs, and a formal proof is not always an adequate justification or explanation of mathematical ideas. Although formal proof has long been thought to produce infallibility in

10 1  Teaching Mathematical Reasoning: A Challenging Task mathematical knowledge, in fact it does not do so (Davis and Hersh 1981; Ernest 1991; Hanna and Jahnke 1996). Standards of rigour are socially constructed (Ernest 1991; Volmink 1990) and “there has never been a single set of universally accepted criteria for the validity of a mathematical proof” (Hanna and Jahnke 1996, p. 884). For example, most mathematics teachers are convinced by the standard one-page presentation of the proof of Pythagoras’ theorem; however, a completely logically rigorous proof would take about eighty pages (De Villiers 1990). Just as in other disciplines, communities of practice (Wenger 1998) exist in the various domains of mathematics, which review new mathematical proofs in accor- dance with the current questions, objects of study, ways of thinking, methods, and results of the specific mathematical domain. The nature of the discipline of math- ematics, founded and built on fundamental, shared concepts means that there is more agreed upon knowledge in mathematics than in other disciplines, such as psychology or sociology. However, this does not mean that mathematical knowl- edge is not socially constructed or contested. Proof does not shield us from the uncertainty of our knowledge (Hanna and Jahnke 1996; Kline 1980). At the same time, proof is an important embodiment of mathematical reasoning and needs to be taught as a particular form of reasoning, justification, and generalization within the discipline of mathematics (Hanna and Jahnke 1996). Creativity and Reasoning A strong rebuttal to the hegemony of proof in mathematics comes from practising mathematicians, who often work intuitively and creatively, searching for under- standing and meaning, rather than rigour and formality. Sternberg and his col- leagues distinguish between creative and analytical thinking (Sternberg 1999; Sternberg et  al. 1998), arguing that “analytical tasks involve analysing, judging, evaluating, comparing and contrasting, and critiquing; creative tasks involve creat- ing, inventing, discovering, imagining and supposing” (1998, p. 374). Although creative and analytical thinking are often posed as dichotomous, they actually sup- port each other in mathematical problem solving and reasoning, for example imag- ining would require some form of comparing and supposing usually requires some analysing. Comparing alternative solutions, ideas, and imaginings all require rea- soning and justification; creative thinking can support links between previously unconnected ideas; and leaps of imagination are often necessary to see a problem from a different perspective. Intuition has also been studied as an important part of mathematical problem solving, creating mathematical arguments, and proving mathematical theorems (Fischbein 1987). Intuition might precede more formal arguments, justifications, and proof, and in some instances, might replace it. A mathematician who intuitively feels that something is wrong in a proof, will search to find the mistake, doubting the proof rather than her intuition (Hanna and Jahnke 1996). Crucial to notions of creativity and intuition is a sense that conviction and understanding do not necessarily

The Centrality of Mathematical Reasoning in Mathematics Education 11 come from formal, deductive, or analytic proofs. Although these have their place, they are certainly not sufficient to solve mathematical problems and communicate mathematical justifications and generalizations. If practices in the mathematics classroom are to be authentic to the discipline of mathematics (Brown et al. 1989), then a broader range of reasoning should be acknowledged and developed in math- ematics classrooms. Empirical and inductive reasoning play an important part in the reasoning practices of mathematicians and mathematics learners, often complementary to theoretical and deductive reasoning. Simon (1996) argues for a notion of “transfor- mational” reasoning, where dynamic transformations of objects are visualised and which provide the reasoner(s) with a sense of conviction and understanding of how and why something is the case. Transformational reasoning supports and is sup- ported by both inductive and deductive reasoning. Drawing on Toulmin, Krummheuer (1995) argues for substantive arguments, rather than merely analytic ones. Substantive arguments show relationships between the main objects and premises, rather than merely drawing deductive conclusions based on previously proved results or axioms. This distinction is similar to Hanna’s characterization of proofs that prove and proofs that explain (Hanna and Jahnke 1996). De Villiers (1990) argues for five key functions for proof – verification, explanation, systematization, discovery, and communication. It is useful to see these as functions of mathematical reasoning as well. Verification establishes that something is the case, i.e. sufficient justification has been produced to confirm that a claim is true. Explanation estab- lishes why something is the case, showing what are the key properties that are necessary for the truth of a claim. Explanatory proofs, or substantive arguments are more satisfying to both mathematicians and mathematics learners (Hanna and Jahnke 1996; Krummheuer 1995). Systematization organizes disparate mathemati- cal concepts that are already established into a coherent mathematical system. As argued above, mathematical reasoning is a key part of mathematical discovery and mathematical reasoning also functions to help communicate our ideas and their warrants to others. The idea that mathematical reasoning involves creativity, discovery, and com- munication is central to the work of this book. In Chap. 4, we show how a col- laborative conversation among learners supported the development of the mathematical concept of function. Communication was the key in enabling learn- ers to make creative, reasoned conceptual leaps. In Chap. 5, we show how the teacher’s practices supported the learners’ mathematical reasoning by encouraging them to question and challenge each other and himself. Again, we see reasoned creativity among his learners. In this section, I have argued that mathematical reasoning is a key element of mathematics and thus is central to learning mathematics in school. I have argued for a broader notion of mathematical reasoning, in which intuition, creativity, imagination, explanation, and communication all play an important role. Fundamental to all forms of mathematical reasoning is the practice of justification and creating adequate arguments in defence of claims. Throughout this section, I have drawn on the notions of mathematical practices, communities of practice, and

12 1  Teaching Mathematical Reasoning: A Challenging Task that mathematics is fundamentally a social practice. In the next section, I explore these ideas further. Theories of Learning and Mathematical Reasoning The work in this book is informed by a number of theories of learning, in particular constructivist, socio-cultural, and situated theories. Following Sfard (1998, 2001), I argue that none of the above theories is sufficient on their own to explain the learn- ing and teaching of mathematical reasoning, and in this project I use them in careful combination. Although some scholars argue that since the fundamental mecha- nisms that generate learning posed by the theories are so different (biological equilibration for constructivists and social relations for socio-cultural and situated theories), the theories may be incommensurable, my argument is that the different mechanisms operate at different levels and in combination with each other and as long as the differences are acknowledged and specified, we can use these theories together to inform teaching and account for learning in mathematics classrooms (see also Sfard 2001). Constructivism Constructivism, in its many varieties, is centrally concerned with how knowledge is constructed and restructured in order to make sense of ever-increasing complex- ity, both in one’s knowledge and in the outside world. Constructivism has had an important influence on theories of mathematics learning and mathematical reason- ing (Confrey and Kazak 2006; Hanna and Jahnke 1996), and on the new curriculum in South Africa (Department of Education 2000). However, just as there are many varieties of constructivism, there are many ways in which constructivism can be misconstrued (Moll 2000). The version of constructivism that informs the work in this book is derived from Piagetian constructivism (Piaget 1964, 1968, 1975), informed by the interpretations of Hatano (1996) and Rowell (1989). Two key principles of this version are first, that what people learn is constrained and afforded by what they know; and second, that there is an integrity to learners’ thinking – what learners think, say, and do makes sense to them in relation to what they know. The role of current knowledge is very particular in that current knowledge is not merely built upon (as in behav- iourist theories); rather it is restructured and reorganized into richer, more con- nected, and more powerful knowledge (Hatano 1996). Just as new knowledge is transformed in relation to prior knowledge, so prior knowledge is transformed in relation to new knowledge. From constructivist perspectives a deepening or trans- forming of thinking involves a deepening or transforming of cognitive structures, either integrating previously separate structures into more general and powerful

Theories of Learning and Mathematical Reasoning 13 structures, or differentiating previous structures into more nuanced structures, which allow for more depth of thinking (Hatano 1996). The implication for math- ematics classrooms is that teachers need to find out how learners are thinking in order to help them build relationships between current and new knowledge. The unit of analysis in constructivist theories is the mind of the individual learner. Social interaction is crucial to constructivism in that it supports and con- strains individual learning (Hatano 1996; Sfard 2001). However, social interaction is a secondary mechanism, important only as long as it engages the key mechanism for learning and development – equilibration (Piaget 1964; Rowell 1989). Equilibration is a biological process where perturbations to current knowledge structures are compensated for in ways that develop them into more powerful structures. Although the initial perturbation might be created by social interaction, the biological pro- cesses must engage for a shift in knowledge to occur. At the same time equilibration on its own is not sufficient to account for learning, because social processes must be taken into account as well. The concept of cognitive conflict explains the links between biological and social processes in constructivism. Cognitive conflict is where a teacher or peer challenges the position of the learner, illuminating a contradiction in her/his think- ing. The theory holds that if the challenge creates a perturbation in the learner, then the learner will equilibrate and develop more powerful knowledge. However, research and experience show that even when learners can see the contradiction, they are often more comfortable maintaining contradictory positions than trying to achieve coherence (Sasman et al. 1998), and might become defensive of their cur- rent knowledge (Balacheff 1991; Chazan and Ball 1999). Although constructivism might be able to account for how people do learn, it is less successful in accounting for how they do not learn (Slonimsky, personal communication). A key part of constructivist research has been work on misconceptions (Confrey 1990; Smith et al. 1993), which, in our experience, has been extremely helpful for teachers. This research shows that learners’ errors are often systematic and consis- tent across time and place, remarkably resistant to instruction, and extremely rea- sonable when viewed from the perspective of how the learner might be thinking. To account for these “rational errors” (Ben-Zeev 1996, 1998) researchers posit the existence of misconceptions, which are underlying conceptual structures that explain why a learner might produce a particular error or set of errors. Misconceptions make sense when understood in relation to the current conceptual system of the learner, which is usually a more limited version of a mature conceptual system (for this reason, many researchers prefer terminology such as “alternate conceptions”). Misconceptions result from structures that apply appropriately in one domain being over-generalized to another, for example, the idea that you cannot take away a big- ger number from a smaller makes sense in the domain of natural numbers, but not in the domain of integers. Thus misconceptions are a normal part of the learning process. Misconceptions have been thought to arise from teaching that emphasizes procedures and individualized instructions (Ben-Zeev 1996; Erlwanger 1975; Schoenfeld 1988). However reports from teachers working conceptually and col- laboratively suggest that misconceptions continue to arise in these classrooms

14 1  Teaching Mathematical Reasoning: A Challenging Task (Ball 1996, 1997; Chazan and Ball 1999; Lampert 2001), which is to be expected, since misconceptions are a normal part of learning. Since misconceptions form part of the learners’ current knowledge, the well established educational truism that teachers need to work with and build on learners’ current knowledge suggests that teachers should work with learners’ errors and misconceptions as well as their correct ideas. Misconceptions alert us to the fact that “building” on current knowledge also means transforming it; current concep- tual structures must change to become more powerful or more applicable to an increased range of situations. At the same time the new structures have their roots in and include earlier limited conceptions (Smith et al. 1993). Learners’ misconcep- tions, when appropriately coordinated with other ideas, can and do provide points of continuity for the restructuring of current knowledge into new knowledge (Carraher 1996; Confrey 1990; Hatano 1996; Smith et al. 1993). The idea that learners’ errors arise from the underlying conceptual structure of the learner and can be an indication of appropriate reasoning and the integrity of the learner’s thinking, can be extremely powerful in helping teachers to shift their teaching towards taking learners’ thinking seriously (Ball and Cohen 1999; Nesher 1987). Teachers who orient toward learner thinking would want to try to understand the thinking that produces the learners’ contributions, including their errors. They would see errors as a normal part of coming to a correct conception. Since miscon- ceptions can also produce correct responses (Nesher 1987), asking learners to explain their thinking when they produce both correct and incorrect contributions is a way to access appropriate or inappropriate underlying mathematical reasoning. In Chap. 8 of this book, I draw on the notion of misconceptions to develop a lan- guage of description for learner contributions, which takes learners’ errors and partial insights seriously while looking for ways to transform them into more appropriate understanding. Socio-Cultural Theories One of the key implications of constructivist theories, which has been popularized in teacher training around the new curriculum in South Africa, is that teachers are “facilitators”, which means that although they might support learning through appropriate tasks and questions, they are not directly implicated in it (Department of Education 1997; Hanna and Jahnke 1996). Teachers are often exhorted not to “tell” learners any mathematics (Chazan and Ball 1999), for fear that they might inhibit the learners’ own constructions. Socio-cultural theories provide a direct challenge to this view, as they argue that adults, and teachers in particular, as bear- ers of the culture, must be involved in developing learners’ understanding and in so-doing, must leave their mark on what learners learn (Hatano 1996; Sfard 2001). The processes of construction will include much of the teachers’ language and ways of seeing that learners appropriate as teachers work with them.

Theories of Learning and Mathematical Reasoning 15 A key difference between socio-cultural and constructivist theories is that socio- cultural theories posit social interaction as the primary mechanism in intellectual development. For Vygotsky (1978), the interpsychological (interaction among people) becomes the intrapsychological (mental functions). Vygotsky has a very strong notion of the mind, which is formed biologically, through the lower mental functions, and socially, through the higher mental functions. He argues that social interaction and broader cultural historical patterns are constitutive of higher order consciousness. The interaction between the social and biological is therefore key, as in Piaget’s constructivism; however, the social is primary. For Vygotsky, social, cultural, and historical knowledge is carried through signs and artefacts and mediated to younger members of the culture by more experienced members. So for Vygotsky, the unit of analysis is always the individual interacting with another person or people, either directly, or through a tool or artefact (for example, a book). This is formalized in his notion of the zone of proximal development (Vygotsky 1978). In some of Vygotsky’s writings, it appears that he conceives of the zone of proximal development as belonging to an individual learner. However, the social nature of his theory suggests that zones of proximal development are created in interaction between learner and teacher or between learner and artefact (Hedegaard 1990; Wertsch 1984). Thus the same teacher/artefact can create different zones of proximal development with different learners, and different teachers/artefacts can create different zones of proximal development for the same learner. Mediation is crucial (Crook 1994; Herrenkohl and Wertsch 1999) in that it creates the conditions of possibility for internalization of the key concepts of the culture. Many teachers, including those whose work is represented in this book, have found the concepts of the zone of proximal development and mediation extremely appealing, because they posit a central role for the teacher. In Chap. 4, we draw on the socio-cultural theory together with constructivism to show how one learner’s development of mathematical reasoning is mediated by conversation with her peers and the teacher, as they use a set of mathematical resources together to solve a problem. We see how the learners’ collaborations are intimately connected with and become part of an individual learner’s increasingly sophisticated reasoning. In Chap. 5, we draw directly on the zone of proximal development, arguing that the teacher’s practices of questioning and listening to the learners’ mathematical rea- soning form a zone of proximal development for the learners and they begin to listen to and question each other, and him, in similar ways. It should be noted here that Vygotsky’s theory is also a theory of social con- struction. Interpsychological processes do not become intrapsychological pro- cesses without being transformed. Vygotsky states: “adults, through their verbal communication with the child, are able to predetermine the path of the develop- ment of generalizations and its final point – a fully formed concept. But the adult cannot pass on to the child his mode of thinking” (1986, p. 120), and Leont’ev writes “the process of internalisation is not transferal of all activity to a pre- existing plane of consciousness; it is the process in which this internal plane

16 1  Teaching Mathematical Reasoning: A Challenging Task is formed.” (1981, p. 57 in Cazden 1988, p. 108). So, internal processes, although constituted by external processes, do not mirror them. There are two strong critiques of Vygsotsky’s work, both relating to his notion of internalization. The first is that the processes of internalization are left relatively un-theorised. This is one area where Vygotsky’s work remains insufficient3 and where Piaget’s notions of assimilation, accommodation, and equilibration do far more to explain how external ideas are internalized. Related to this is Vygotsky’s under-acknowledgment of the role of errors and misconceptions in learning. His work often suggests that learning proceeds relatively smoothly in the zone of proxi- mal development. The second critique is that there is far too much emphasis on internalization in Vygotsky’s theory, because of its strong focus on mind (Crook 1994; Daniels 2001; Lave 1993). This critique suggests that a focus on social rela- tions is more useful in understanding learning. Related to this argument is the fact that Vygotsky’s theory, while acknowledging necessary asymmetrical relationships between teacher and learner, does not always acknowledge the power differences among learners, particularly in relation to race, gender, and class and how these might affect learning. A more participatory account allows for these to be included. Situated Theories Situated theories view learning as participation in communities of practice (Lave 1993; Lave and Wenger 1991; Wenger 1998). To view learning as participation is to say that not only does learning occur through participation, as both constructivist and socio-cultural theories argue, but that learning is defined and identified as increasing participation in a practice. To learn is to participate better. To learn math- ematics is to become a better participant in a mathematical community and its practices, using the discursive tools and resources that the community provides (Forman and Ansell 2002; Greeno and MMAP 1998). The unit of analysis in situ- ated theories is the community of practice and it is important to specify the prac- tices of a particular community (Brown et  al. 1989). Communities of practice constitute contexts for the learning of their members, and as communities they also learn (Wenger 1998). The mechanism for learning in situated theories is legitimate peripheral participation in communities of practice (Lave and Wenger 1991). Newcomers to the practice participate legitimately, but on the periphery, at first. As they gain experience, their participation shifts towards full participation, which is their learning. As newcomers become oldtimers, so the community itself learns and shifts, creating both personal and communal growth. The processes of negotiation between newcomers and oldtimers can create tensions and conflict, as newcomers 3 I thank Steve Lerman for pointing out that this is probably because Vygotsky died a short time after formulating the notion of zpd and internalization.

Theories of Learning and Mathematical Reasoning 17 stake their positions in the community. Thus power relations, which are not taken into account by constructivist and socio-cultural theories, become the key to learning in situated theories. Lave and Wenger argue that their theory of social practice “emphasizes the inherently socially negotiated character of meaning and the inter- ested, concerned character of the thought and action of persons-in-activity” (Lave and Wenger 1991, p. 50). Similar to socio-cultural theories, situated theories view the role of the social interaction as constitutive of learning; they are social theories of learning (Wenger 1998). Different from socio-cultural theories, they view learning as only a social phenomenon; the definition of learning as participation rather than constituted by participation suggests learning is social and not mental. Learning is the creation of identities in communities of practice (Wenger 1998). Situated perspectives argue that knowledge cannot be seen as the “possession” of an individual, but rather is distributed among people and resources (Greeno et al. 1996; Sfard 1998). In situated perspectives, a concern with thinking is transformed into a concern with participation, with how learners use mathematical tools and discourse to rea- son and justify their reasoning (Sfard 2001). Making connections and generalizing ideas are important in situated perspectives, however the connections and general- izations are ideas that are in the conversation, rather than structures in the head. Greeno and MMAP (1998) suggest that situated analyses broaden the notion of conceptual structures into one of attunement to affordances and constraints. Affordances and constraints are located in interactional situations, in the classroom, the domain of mathematics, and the lives of the learners beyond school. From this perspective, learners who do well in mathematics do so because they align and identify with the requirements and expectations of the classroom, both mathemati- cal and social. A learner may struggle to learn, not because she has not developed appropriate conceptual structures, but because she is responding to a different task than the one set by the teacher, or does not want to be seen to be too intelligent or not intelligent enough in front of her peers, or has decided not to engage with math- ematics because it does not seem to be important in the lives of people who are important to her. Such attunements are patterned regularities, which may be just as important in accounting for learning as are conceptual structures. The de-emphasis of conceptual structures as products of learning makes situ- ated theories somewhat difficult for teachers to own and work with, epecially teachers, like the teachers in this book, for whom conceptual understanding and mathematical reasoning are important. However, there is one key notion in situated theories – communities of practice – which attract teachers, and which seem to be compatible with the key elements of socio-cultural and constructivist theories. Communities of practice present images of how communication can take place in classrooms, the roles of resources, of different learners, and of the teacher. Although classroom communities must be somewhat different from communities outside of classrooms (Lave 1993, 1996; Lerman 1998), classroom communities, where genuine mathematical communication and the development of mathematical understanding and identities take place, can be established (Boaler 2004; Boaler and Greeno 2000; Boaler and Humphreys 2005; Lampert 2001; Staples 2004).

18 1  Teaching Mathematical Reasoning: A Challenging Task This view infuses the work in Chaps. 4–6 of this book, where we show how communities of practice were created in three of the classrooms. It also informs Chaps. 8 and 9, which show how learner contributions and teacher responses c­ o-produce each other, and Chaps. 10 and 11, where dilemmas of teaching and resistance to teacher change are viewed as profoundly situated and developed in and through communities of practice. Teaching Mathematical Reasoning The theories discussed above are primarily theories of learning. It is often thought that theories of learning have direct application to classrooms and suggest particu- lar pedagogical approaches. However, this is not the case; rather theories of learn- ing suggest general pedagogical principles and implications for pedagogy; they do not directly lead to particular pedagogical approaches. Moreover, pedagogical prin- ciples do not derive from theories of learning in a one-to-one correspondence. Different theories might suggest very similar approaches, which are distinguished at the level of explanation rather than at the level of practice. It might be tempting to conclude that since constructivist theories focus on the individual, they suggest individual approaches to teaching and learning, while socio-cultural and situated perspectives suggest group work. However, all three theories suggest that group work is a useful pedagogical approach and none would advocate that learners do no work on their own. From all three perspectives, encouraging learners to talk through their ideas with each other is an important process, as is encouraging learners to write down different versions of their thinking, for themselves and others. Constructivist perspectives suggest that when learners are pushed by others to articulate their thinking, they are likely to clarify their thinking, both for others and for themselves (Barnes and Todd 1977; Glachan and Light 1982; Mercer 1995; Vygotsky 1986). In the process of clarifying their thinking, learners might develop more complex concepts, through differentiation, integration, and restructuring (Hatano 1996). Situated perspectives suggest that as learners consider, question, and add to each other’s thinking, important mathematical ideas and connections can be co-produced. For constructivist perspectives the group is a social influence on the individual; for sociocultural and situated perspectives the group is the important unit, which produces mathematical ideas within or beyond the individual learner. One, or both of these purposes for group work might be operating in a classroom at any particular time. In this section, I delineate pedagogical implications for teaching mathematical reasoning, drawing on the arguments in the previous sections of this chapter. The key in teaching mathematical reasoning, as in teaching any other aspect of mathe- matical proficiency, are the kinds of tasks that learners engage in, the ways in which they engage with these tasks, and the kinds of interactions around the tasks among the learners and the teacher. However, as noted by Ball and Bass, “simply posing open-ended mathematical problems that require mathematical reasoning is not

Teaching Mathematical Reasoning 19 ­sufficient to help students learn to reason mathematically. Neither is merely asking students to explain their thinking” (2003, p. 42). Tasks for Mathematical Reasoning A number of frameworks have been developed to describe the complexity of tasks (Biggs and Collis 1982; Shavelson et al. 2002; Stein et al. 1996, 2000). In this project we drew mainly on the work of Stein et al. (1996, 2000). Although we acknowledge limitations with this framework (Sanni 2008a), it was very use- ful as a starting point for teachers wanting to select tasks to support learners’ mathematical reasoning. This framework is discussed in detail in Chap. 3, where we use it to analyse how learners responded to tasks intended to develop math- ematical reasoning. For the purposes of this chapter it is sufficient to note that Stein et  al. identify task features which support higher cognitive demands on learners, including reasoning and sense making. These features are “the exis- tence of multiple-solution strategies, the extent to which the task lends itself to multiple representations, and the extent to which the task demands explanations and/or justifications from the students” (Stein et al. 1996, p. 461). Stein and oth- ers (Ball 1993; Boaler and Humphreys 2005; Chazan 2000; Lampert 2001) show that tasks that support multiple voices, disagreements, and challenges also sup- port mathematical reasoning, when used appropriately. Douek (2002) argues for specific kinds of complexity in tasks to support the development of mathemati- cal arguments, including the complexity of integrating a number of different arguments into a coherent whole, the complexities involved in moving from dynamic to static representations, and the complexity of the contexts in which tasks are set. Garuti and Boero (2002) describe teaching experiments where the arguments of famous scholars (Galileo, Plato) are presented to students as examples of forms of argument, and students are asked to write similar argu- ments for mathematical problems. Considering the arguments of others, includ- ing those of one’s own peers, can be a powerful source of developing a learner’s own reasoning and arguments. Choosing appropriate tasks is necessary but not sufficient to support a learner to develop reasoning. Stein et al. (1996) show that, with support, the teachers in their project chose tasks that made higher order cognitive demands on learners. However, as the tasks were implemented in the classrooms, the level of demand declined. In South Africa, Modau and Brodie (2008) show how a teacher teaching the new cur- riculum in Grade 10, supported by a new curriculum textbook, chose tasks that required reasoning. However, at implementation, he was not able to maintain the level of the tasks, but through his questioning and patterns of interaction, lowered the task demands and thus did not support reasoning (Jina and Brodie 2008). Sanni has shown that in six Nigerian classrooms the level of most of the tasks also declined. However, when he worked as a support for one of the teachers, the level of the tasks remained high and the learners’ reasoning improved (Sanni 2008b).

20 1  Teaching Mathematical Reasoning: A Challenging Task Taken together these studies suggest that substantial work with teachers is required to support them to interact with their learners on tasks to support the learners’ mathematical reasoning. Classroom Interaction Since communication is fundamental to reasoning (Ball and Bass 2003; Douek 2005; Krummheuer 1995), it makes sense that learners should discuss their reason- ing with others. This is supported by all the learning theories discussed previously and by curriculum reforms in South Africa and internationally. As learners attempt to create reasoned arguments for their ideas, they help themselves and each other to clarify their thinking and they are able to create some of the practices that math- ematicians engage in as they produce arguments and justifications. However com- munication can be structured in a variety of ways, leading to very different kinds of support for mathematical reasoning. Many South African teachers talk about the “question and answer method” as if this guarantees learner participation. However, it is well known in the research literature that if the questions are narrow and do not challenge learners’ thinking, then the resulting interaction is stilted and does not support reasoning (Bauersfeld 1988; Edwards and Mercer 1987; Mehan 1979; Nystrand and Gamoran 1991). On the other hand, putting learners into groups and leaving them to work without mediation from the teacher does not necessarily pro- vide enough support for developing their reasoning (Brodie and Pournara 2005). Even whole class discussions are often not successful, because working out exactly how to respond to the learners’ developing ideas and reasoning is a difficult task for teachers (Heaton 2000). The work on classroom interaction in Chaps. 8–10 of this book draws on work done many years ago by Mehan (1979) and Sinclair and Coulthard (1975) on the pervasive exchange structure of classroom discourse, the Initiation-Response- Feedback/Evaluation (IRF/E) exchange structure. The teacher makes an initiation move, a learner responds, the teacher provides feedback or evaluates the learner response and then moves on to a new initiation. Often, the feedback/evaluation and subsequent initiation moves are combined into one turn, and sometimes the feed- back/evaluation is absent or implicit. This gives rise to an extended sequence of initiation-response pairs, where the repeated initiation works to achieve the response the teacher is looking for. When this response is achieved, the teacher positively evaluates the response and the extended sequence ends. Neither Sinclair and Coulthard nor Mehan evaluated the consequences of the IRE structure. Other researchers (Edwards and Mercer 1987; Nystrand et al. 1997; Wells 1999) have argued that it may have both positive and negative consequences for learning. Although this structure requires a learner contribution every other turn (the response move), and therefore apparently gives the learners time to talk, much research has shown that because teachers tend to ask questions to which they already know the answers (Edwards and Mercer 1987) and to “funnel” learners’

Teaching Mathematical Reasoning 21 responses toward the answers that they want (Bauersfeld 1988), space for genuine learner contributions is limited. At the same time, it is very difficult for teachers to move away from this structure (Wells 1999) and so, in trying to understand a range of pedagogies, it is important to try to understand the benefits that it affords. Whether the IRE has positive or negative consequences for learning will most likely depend on the nature of the elicitation and evaluation moves, which in turn influ- ence the depth and extent of the learners’ responses. In Chap. 9, I develop a lan- guage to describe teacher responsiveness in the fused elicitation/evaluation moves, which distinguishes a number of teacher moves and their consequences for learner contributions. One aspect of classroom interaction that has been identified as important to sup- porting useful interaction, is the development of ground rules (Edwards and Mercer 1987) or classroom norms (McClain and Cobb 2001), which are different from those in traditional classrooms. Norms that support reasoning would includethe following: learners are called on to justify all their reasoning, not only mistaken reasoning as might happen in traditional classrooms; learners are expected to listen to and build on each others’ ideas and challenge them where necessary; learners can and should challenge the teacher, and the teacher should justify her/his mathemati- cal thinking. This raises the important issue of authority in mathematics class- rooms. Traditionally, mathematics learners are expected to accept the authority of the teacher or the textbook, rather than the authority of mathematical justifications. These two kinds of authority are very different (Brousseau and Gibel 2005), and are implicated in the learners developing a productive disposition towards mathematics (Kilpatrick et  al. 2001) and hence achieving overall mathematical proficiency. Productive disposition is a belief that mathematics can and does make sense, and that every learner can make sense of it. This requires an understanding that the “rules” are not arbitrarily decided on by powerful individuals, but that they make sense in terms of a system of knowledge, which can be understood by everyone, with sufficient effort. Ball and Bass (2003) argue that mathematical justification is grounded in a body of public mathematics knowledge, where this knowledge can be that of a group of mathematicians, or an elementary classroom community. This public knowledge ensures that individual sense making becomes accountable to a broader commu- nity; because an idea making sense to an individual is not the same as an idea being based on shared reasoning in communities of mathematicians over time. So taking the individual learner’s reasoning seriously means attempting to connect it to accepted mathematical reasoning. They argue, “reasoning, as we use it, comprises a set of practices and norms that are collective, not merely individual or idiosyn- cratic, and rooted in the discipline” (p. 29). It follows from their position, although they do not argue it, that even the classroom mathematics community cannot be the final arbiter on the acceptance of a mathematical argument, because this commu- nity is accountable to broader communities of mathematical practice and to the discipline of mathematics. The idea of accountability to the discipline is one that attracted all of us in this project, and subsequent chapters will show how we worked with this notion and how the learners received it.

22 1  Teaching Mathematical Reasoning: A Challenging Task The Challenges of Teaching Mathematical Reasoning In this chapter, I have outlined the theoretical positions on mathematics, learning and teaching, which inform this book and have suggested some of the many chal- lenges that teachers face when trying to make reasoning a central part of their practice. I have also suggested some challenges that researchers and teacher educa- tors face when looking for ways to talk about the teaching and learning of mathe- matical reasoning. In the rest of this book, we take up some of these challenges: • How do learners respond to tasks chosen to elicit their mathematical reasoning? • How can teachers interact with learners around tasks to engage their mathemati- cal reasoning? • How can teachers teach to develop mathematical proficiency? • How does collaborative conversation among learners and teacher promote math- ematical reasoning? • What kinds of teaching practices, questions, and moves help to encourage and sustain the learners’ mathematical reasoning? • How can we describe the learners’ contributions and teachers’ responses to these in ways that can help us talk about them in more specific ways? • What kinds of dilemmas do teachers experience as they teach mathematical reasoning? • What can teachers do in response to the resistance to new ways of teaching? We approach these questions from two directions. In part two of this book, we present five studies conducted by teachers in their classrooms, each of which addresses one or two of these questions. The range of contexts in which the teachers work and their approaches to the topic of mathematical reasoning suggest a number of possibilities to other teachers, teacher educators, and researchers wanting to undertake case-studies of teachers teaching mathematical reasoning. In part three of the book, I look across the teachers’ practices, in a multiple-case study. I suggest a language of description for learner contributions and teacher moves, and I use this language to illuminate ways in which teachers, teacher educators, and researchers can gain deeper insight into how to respond to the learners’ mathematical thinking. I also use this language to illuminate some of the dilemmas that teachers experience when engaging learners’ mathematical reasoning and to talk about the resistance that they may experience. It is well established that meaningful change in teaching and learning takes time. In this book we illuminate both successes and challenges in teaching for mathemat- ical reasoning, among ordinary teachers, to give substance to why such teaching takes time to achieve. We do not claim to have succeeded in producing the ideal teaching and learning situations that reformers might hope for, and we are not sure whether such perfection is possible. We do claim to have learned much about what we have achieved and how we might move forward. We hope that our work in developing research and practice together will provide ideas and possibilities for many other teachers, teacher-educators, and researchers to begin their own explora- tions in teaching mathematical reasoning.

Chapter 2 Contexts, Resources, and Reform In the previous chapter, I outlined possibilities for teaching mathematical reasoning that involves learners communicating their thinking to their teacher and their peers and teachers taking learners’ mathematical reasoning seriously to develop and transform it. This is in line with the visions of reform mathematics in a number of countries. However, international evidence suggests that very few teachers embrace reforms and those who do, experience significant challenges in their teaching. The challenges that I outlined at the end of the previous chapter are daunting in any context even in the most well resourced contexts. However, in South Africa and many other countries, resources in most schools are severely limited, adding to teachers’ difficulties in enacting reforms. At the same time, resources are not the only influence on reform teaching and the ways in which they exert an influence are not always obvious. In a review of recent international and South African studies, Fleisch (2007) shows that the studies are inconclusive on the effects of resources such as teacher qualifications, class size, and learning materials on learner achievement. This suggests that there are ways in which resources that do matter are most likely mediated by other variables. In this chapter, I discuss some responses to reform pedagogy across a range of contexts and discuss ways in which contextual constraints and resources may or may not be implicated in enactments of reform teaching. This discussion, together with a more general discussion of the resources available for teaching and learning in South Africa, serves to situate the description of the different contexts of the teachers in this study and the resources available to them as they worked to teach mathematical reasoning in their classrooms. Responses to Reforms A strong impetus for reform curricula in many countries is the need to redress inequalities in mathematics education. Internationally, success in mathematics is distributed according to race and socio-economic status (Association for Mathematics Education of South Africa 2000; Department of Education 2001; K. Brodie, Teaching Mathematical Reasoning in Secondary School Classrooms, 23 DOI 10.1007/978-0-387-09742-8_2, © Springer Science+Business Media, LLC 2010

24 2  Contexts, Resources, and Reform Moses and Cobb 2001; Secada 1992). While many of the reasons for this maldis- tribution originate outside of the classroom, there are arguments that classroom practices can begin to work towards equity (Boaler 2002). Allowing different ways of knowing mathematics to be available in the classroom may afford success for a wider range of learners (Boaler 2002; Boaler and Greeno 2000). Allowing learners to express their ideas in the classroom can lend to diverse ways of thinking, and help to teach learners that everyone’s thinking can contribute to the development of mathematical knowledge (Lampert 2001). There has been much debate as to whether current mathematics reforms can be a mechanism for ensuring more equitable participation and achievement in mathe- matics (see Brodie 2006, for a summary of these debates). Empirical evidence in well-resourced countries is beginning to show that reforms do mitigate achievement gaps between marginalised and other learners and also enable learners to develop more motivated and positive identities as mathematics learners (Boaler 1997; Boaler and Greeno 2000; Hayes et al. 2006; Kitchen et al. 2007; Schoenfeld 2002). However, the evidence also shows that implementation of reform curricula is not widespread and in fact it is likely that implementation of reforms is inequitably distributed (Kitchen et al. 2007), so that poorer learners are less-likely to experience reform curricula and pedagogy. Particularly in African contexts, issues of resources, including big classes and few materials, teacher confidence and knowledge, and support for teachers, can be major barriers to developing new ways of teaching (Tabulawa 1998; Tatto 1999). If reforms are successful in promoting equity and if they are not taken up in less-resourced contexts, then the existing division between rich and poor are likely to be exacerbated. There is also growing evidence that teaching in reform-oriented ways is an extremely challenging task for teachers (Sherin 2002; Nathan and Knuth 2003) and that successful reform teachers are rare, even in well resourced schools in the United States where the reforms have been in place for 10 years longer than in South Africa. Among the 18 teachers in their study, Fraivillig et  al. (1999) considered only six teachers to be “skillful” in eliciting and supporting learner thinking, while only one was successful in eliciting, supporting, and extending learner thinking. Hufferd- Ackles et  al. (2004) described the development of reform practices through four levels. Of the four teachers they worked with, only one teacher’s trajectory took her and her learners through all four levels. These studies were conducted in well- resourced classrooms and so suggest that while resources may be important, they are not the only challenge for teachers in working with learners’ reasoning. Studies of and by teachers who are successful in developing discussion and col- laboration around learners’ reasoning, identify a number of challenges in such work. These include: supporting learners to make contributions that are productive of further reasoning (Heaton 2000; Staples 2004); respecting and valuing all learn- ers’ thinking while working with the diversity of their mathematical ideas (Lampert 2001); respecting the integrity of learners’ errors while trying to transform them and teach the appropriate mathematics (Chazan and Ball 1999); seeing beyond one’s own long-held and taken-for-granted mathematical assumptions in order to hear and work with learners’ reasoning (Chazan 2000; Heaton 2000); maintaining a “common ground”, which enables all learners to follow the conversation and its

Responses to Reforms 25 mathematical purpose and to contribute appropriately (Staples 2004); and generating mathematical practices such as making connections, generalizing, and justifying (Boaler and Humphreys 2005). The above research shows that the pedagogical demands of mathematical conversations can be daunting and that we need to under- stand more about the practices involved in generating and sustaining these conver- sations (Brodie 2007b). Research on the new curriculum in South Africa has shown that teachers who are enthusiastic about and express support for the new curriculum struggle to enact many of the ideas in their classrooms. These studies show that many classrooms remain teacher-centred, and teachers engage with learners’ ideas in superficial ways, if they do so at all (Chisholm et al. 2000; Taylor and Vinjevold 1999). Other studies show some hybrid practices beginning to develop. Jansen (1999) found that while most teachers were not implementing the new curriculum, or were doing so very superficially, some of the more experienced and confident teachers were able to move between old and new practices and negotiate for themselves and with their learners what it means to implement the new curriculum. Brodie et al. (2002) found that many teachers set up tasks and group work situations where learners engaged with the tasks. However, when learners expressed their thinking, teachers struggled to support and engage with their reasoning to take them further and to develop them mathematically (Brodie 1999; Brodie et al. 2002). This finding was confirmed in more recent studies, where teachers selected tasks that could elicit mathematical reasoning, but did not engage learners’ reasoning in classroom interaction (Jina and Brodie 2008; Modau and Brodie 2008; Stein et al. 1996; Stein et al. 2000). There are a number of possible explanations for South African teachers’ difficul- ties with the new curriculum. One claim, prominent at the moment, is that teachers do not know enough conceptual mathematics to teach in ways required by the new curriculum (Taylor and Vinjevold 1999). Other explanations are that teacher devel- opment around the new curriculum has been inadequate and that appropriate cur- riculum materials are not available (Chisholm et  al. 2000; Taylor and Vinjevold 1999). A third possibility is that teachers are able to implement some aspects of the new curriculum, for example, higher level tasks, relatively easily, whereas other aspects, in particular, interaction with learners, are particularly difficult (Brodie et  al. 2007). Slonimsky and Brodie (2006) argue that new curriculum practices require that teachers coordinate a complex set of contextual and knowledge con- straints and that such coordination takes a long time to develop. All the above explanations acknowledge that resources are only part of the problem. One of the aims of this project was to explore possibilities for developing learn- ers’ mathematical reasoning in a range of South African contexts and thereby begin to develop a deeper understanding of how resources are implicated in such prac- tices. The next section presents some background on educational resources and achievement in South Africa and the following section describes the differently resourced contexts in which the five teachers in the project worked. In the discus- sion of resources, I include the material resources of the schools, the human resources, in particular teacher and learner knowledge, and finally the resources that the teachers chose to work with in the study – the tasks that they developed to engage their learners in reasoning mathematically.

26 2  Contexts, Resources, and Reform The South African Context As with all aspects of life in South Africa, the education system is characterized by large disparities between rich and poor, and most of our schools and learners are of very low socio-economic status. Most teachers in South Africa teach big classes in very poorly resourced schools. The latest national data shows that of the 25,145 operational schools in South Africa, 11.5% of schools did not have water, 16% had no electricity, 5.2% did not have ablution facilities, 80% did not have libraries and 67% did not have computers for teaching and learning (Department of Education 2007). The average learner–teacher ratio in schools was 32:1 and the average learner-classroom ratio was 38:1 (Department of Education 2008). These averages hide wide disparities between provinces and schools. A research study conducted in Gauteng and Limpopo (the richest and poorest provinces in South Africa respectively), observed averages of 35 students per class in secondary schools, with the three rural secondary schools averaging 60 learners per class and with some classes having as many as 120 learners (Adler and Reed 2002). There were limited resources such as overhead projectors and those resources that did exist, for example chalkboards, were often in poor condition. While outrageous in any terms, this lack of resources is particularly significant for teachers attempting to work with their learners’ mathematical reasoning. It is difficult to attend to learners’ ideas when there are 50 or more learners in the class and few mate- rial resources. Moreover, many learners, having experienced poorly resourced educa- tion, often have weak mathematical knowledge (Fleisch 2007), and may be reluctant to participate in lessons. When they do participate, they may express barely coherent, or very problematic ideas, and teachers may not be able to engage with these ideas (Brodie 2000). The fact that most teaching and learning takes place in English, which is not the main language of most teachers and learners, also makes participation more difficult for learners and development of learner thinking more difficult for teachers. Learners’ weak mathematical knowledge is apparent in the annual results of the school leaving examinations1, which are taken by approximately 500,000 Grade 12 learners each year, of which about 300,000 take the mathematics examinations. In 2005, 55% of about 303,000 learners passed mathematics and in 2006, 52% of about 317,500 learners passed mathematics. In 2005, about 44,000 learners took the examination at the Higher Grade level, which is required for entry into scientific fields at university, and 59% passed while in 2006, about 47,000 took the examina- tion at this level and 53% passed (Department of Education 2008). The inequities of the system become apparent when we see that of 40,000 Higher Grade candi- dates in 2001, only 20,000 were “African”,2 and of these about 3,000 passed. Thus, 1 South Africa has school learners examinations at the end of Grade 12. These examinations are high stakes and determine students’ eligibility for further study and job opportunities. 2 The apartheid system of racial classification was four-tiered and funding for education was directly linked to these tiers. Although many white South Africans refer to themselves as African, in this context “African” refers to black South Africans who are not “coloured” or of Indian descent, and whose schools and colleges received least funding under apartheid. Black Africans make up about 80% of the South African population.

The South African Context 27 while about 85% of white Higher Grade candidates passed, only 15% of black African candidates did. Figures for 2004 are that 7,236 African learners passed out of a total of 40,000 candidates3 and of these, 2,406 African learners passed with a “C” grade or higher, which is 0.5% of the total number who wrote mathematics and 6% of those who wrote Higher Grade mathematics (Centre for Development and Enterprise 2007). Research in the lower grades shows that learners begin to struggle with mathe- matics as early as Grade 3. Contrary to other developing countries, South African learners are almost all in school. Fleisch et al. (2008) show that in 2007, more than 95% of children of compulsory school age attended an educational institution and that this reflects an improvement since 2001 in each age cohort between 7 and 15 years of age. However, Motala and Dieltiens (2008) raise questions as to what these learners actually learn in school, suggesting that about 60% are disengaged and disaffected and learn very little. Reviewing the research in mathematics, Taylor et al. (2003) conclude that “studies conducted in South Africa from 1998 to 2002 suggest that learners’ scores are far below what is expected at all levels of the schooling system, both in relation to other countries (including other developing countries) and in relation to the expectation of the South African curriculum.” Many Grade 3 learners struggle with basic skills such as adding and subtracting two-digit numbers that require “carrying” or “bor- rowing”. South Africa also performs poorly in international comparison studies. In the third TIMSS study, South Africa came last out of 41 countries at all three grade levels tested, doing significantly worse than countries with similar GDPs, for exam- ple Latvia and Lithuania (Howie and Hughes 1998). In the 2003 TIMSS study, South African Grade 8 learners again came last out of 46 countries, and more sig- nificantly did worse than countries with lower Human Development Indices, includ- ing Ghana et al. (Reddy 2006). In the Monitoring Learner Assessment Study (MLA), which compared Grade 4 learners across 12 African countries, South Africa had a mean score of 30% for numeracy, which was the lowest of all 12 countries (Taylor et  al. 2003). These average scores hide the large disparities between black, low socio-economic status learners and wealthier, white learners, but they serve to show the extent of the “crisis” in mathematics learning in South Africa, which is quanti- tatively different from many other countries. While we acknowledge critiques and limitations of such comparative studies (for example, Keitel 2000; Reddy 2006), they do serve as an indication of some of the challenges that our education system faces and the difficult conditions under which many teachers work. The “crisis” also extends to the availability and quality of mathematics teachers in South Africa. Many South African teachers, because they were under-served by apartheid education, have relatively weak knowledge of mathematics and how best to teach it (Taylor and Vinjevold 1999). This situation does not look set to change in the near future. Given the limited numbers of students who graduate from school with strong mathematical knowledge, the pool for potentially well-qualified 3 The number of African candidates who wrote is not available.

28 2  Contexts, Resources, and Reform teachers is small. Students who do well in mathematics and science usually have a range of more attractive career options in other fields. Knowledgeable teachers of mathematics are often recruited by industry with far better salaries and working conditions. There is a lack of detailed data about mathematics teachers in South Africa (Centre for Development and Enterprise 2007) but the following give only a part of the picture over the past 10 years. In 1997, only 50% of teachers of math- ematics had specialized in mathematics in their training (Department of Education 2001); in 2004, a survey of 1,766 secondary schools (out of a total of about 5,600) showed that there were 1,734 qualified mathematics teachers at these schools and of those only 1,362 were actually teaching the subject (Centre for Development and Enterprise 2007); and in 2006, 16 universities graduated a total of about 550 math- ematics teachers (Centre for Development and Enterprise 2007). I have taken some time to review these statistics because they provide a back- ground for understanding the debates about the new curriculum in South Africa, and the contexts of the schools in this study. It is imperative to provide access to mathematics for large numbers of low socio-economic status learners. The govern- ment’s response has been the development of policies that encourage the transfor- mation of the curriculum and pedagogy in South African schools. The curriculum was developed in consultation with local and international experts and draws on the international research that suggests that reform pedagogy can reduce inequality in mathematics achievement. However, the international experience of teacher diffi- culties with reform curricula, in addition to the particular challenges of the South African situation suggest that we cannot assume that the new curriculum will reduce inequality – it may even increase it. This was a concern for all of the teachers in this project, and so we wanted to examine how teachers worked with learners’ mathematical reasoning in a range of South African classrooms. Five Schools: Contexts and Resources Race and Socio-Economic Status Fifteen years after the end of apartheid, although there have been some shifts, schools still largely reflect historical divisions of race and class. Table 2.1 gives a description of each of the five teachers’ schools in terms of race and socio- economic status4. 4 Historically race and class have been closely related in South Africa. Although this is changing for some small sections of the population, to a large extent this trend still exists and is largely reflected in this sample. In order to determine socio-economic status, we used the location of the school, the school fees, which public schools in South Africa are allowed to charge, and the teach- ers’ knowledge of the typical occupations of the parents.

Five Schools: Contexts and Resources 29 Table 2.1  Demographics of schools Learners’ socio-economic School fees No. of teachers/ (per year) no. of learners Teacher Learners’ racea status Mr. Nkomo Black Working class R200 1,650/42 (39:1) Mr. Mogale Black Working class R200 1,700/46 (37:1) Mr. Daniels All races Middle and lower-middle R4,000 1,600/60 (27:1) Mr. Peters Black and class R400 1,250/36 (35:1) “coloured” Working class Ms. King R40,000 850/65 (13:1) White, with a Middle and upper-middle few learners class (private) of other races aWe use apartheid terminology to describe race, which is standard practice in South Africa to indicate shifts or lack thereof in historical racial divisions Under apartheid Mr. Nkomo’s and Mr. Mogale’s schools served only black learners, Mr. Peters’ school served only “coloured” learners, and Mr. Daniels’ and Ms. King’s schools served only white learners. Ms. King’s school is a private, boys- only school, with some boys living at the school, and all the others are public, co- educational non-residential schools. The racial profile of the teachers matched those of the learners. Since schools began to integrate in the early 1990s, Mr. Peters’ school has black and “coloured” learners; Mr. Daniels’ school is racially diverse with learners from all four racial “groups”; and Ms. King’s school has a few black, “coloured”, and Indian learners, but is still predominantly white. Teacher diversity across the schools has occurred much more slowly. All the teachers in Mr. Nkomo’s and Mr. Mogale’s schools are black, most of the teachers in Mr. Peters’ school are “coloured”, with some black teachers, almost all of the teachers in Ms. King’s school are white, with a few teachers of other races. Mr. Daniels’ school has made the most progress in integrating teachers across race. Although many teachers are still white, there are a number of “coloured”, Indian and black teachers at the school. Mr. Daniels’ himself is “coloured”, and moved to this school from a “coloured” school about 4 years ago. School Resources All of the five schools are known in their areas as good schools. They all regularly achieved pass rates of 65% and above in the Grade 12 examinations, well above average nationally, and a little above average for Gauteng province in which they are located (Motala and Perry 2002). They are all functional most of the time, in contrast to many other schools in South Africa (Christie and Potterton 1997; Taylor et  al. 2003; Taylor and Vinjevold 1999). This means that school starts on time, most learners are present, absentees are noted, learners move between classes relatively quickly, teachers come to class and teach, learners return to classes after breaks, there are

30 2  Contexts, Resources, and Reform regular teacher meetings, and there are administrative staff and administrative c­ omputer systems. According to the teachers, the principals are supportive of efforts to improve teaching and learning in their schools. All the principals supported the teachers studying further and all were eager for this research to take place in their schools. So, while the five schools represent some diversity, they do not capture the full diversity of South African schools. The study is limited to an urban area, in the ­wealthiest province in South Africa, and well-functioning schools. However, as the first study of teaching mathematical reasoning in South African high schools, it was important and appropriate to limit our study in this way. All of the five schools have fences or walls and control access to the schools5. Ms. King’s school is located on a beautiful, large, peaceful property, situated next to a busy business district. There are dormitories for the learners who live on cam- pus, houses for the teachers who live on campus, a church, plenty of sports fields and a dam with guinea fowl. It is easy to forget that you are in the middle of a big city while at the school. Mr. Daniels’ school is located in a residential suburb, on a hill with a beautiful view of neighbouring suburbs. It has a number of sports fields and has a “green” feel to it. Mr. Nkomo’s, Mr. Mogale’s, and Mr. Peters’ schools are located in residential areas which are poorer and less “green”. There is little open space in Mr. Peters’ school and some dusty fields in the other two schools. Mr. Peters’ school is an area that is well known for gang activity and violence. Sometimes learners come to school with knives and there have been some incidents with guns. During the time of the study, a teacher was robbed at gunpoint of his laptop and cellphone in the school grounds, an incident that caused considerable disruption to teaching and learning for about a week. Learners also are subject to attack when they leave the school, especially many black learners who come from other areas and have to walk through “coloured” gang territory in order to get home. Table  2.2 indicates other resources available at each of the teacher’s schools. Table 2.2  Resources available at the schools Teacher Staffroom Library Computer room Photocopying Non functional Yes Mr. Nkomo Yes Old books: Yes Mr. Mogale Yes mainly: Yes textbooks: Non functional Yes Yes Yes Mr. Daniels Tea and coffee Well equipped Yes Mr. Peters Yes No Ms. King Tea, coffee, Well equipped computers 5 Christie and Potterton (1997) argue that this contributes to the functionality of schools.

Five Schools: Contexts and Resources 31 Classroom Resources Each teacher chose one Grade 10 or 11 class in which we would conduct the research. Table  2.3 gives an overview of the classes that comprised the research sample. The above table shows the disparities in class size among the teachers, in rela- tion to the socio-economic status of learners at the schools. The schools of lower socio-economic status tend to have larger classes. The only class that does not fit the trend is Mr. Nkomo’s, where mathematics classes are smaller in Grade 11 and 12. There is a difference of almost 20 learners between Mr. Peters’ and Ms. King’s classes. The levels of the classes relate both to tracking practices at the schools and the level of examination that learners are being prepared for. Some schools teach standard and higher grade learners in the same class (Mr. Nkomo’s school in this study) while some differentiate them (Mr Mogale’s and Mr. Peters’ schools). Some schools track even further beyond this (Mr. Mogale’s and Ms. King’s schools). Ms. King’s classroom is part of a newly built wing of the school, is carpeted and has air-conditioning. There is a big table and chair for each learner, which can be arranged for work in groups. There are whiteboards and pens, a teacher’s desk with a computer, cupboards and tables for storing paper and worksheets, notice boards filled with math posters, an overhead projector and screen, and a television set which can be used for presentations from a computer. Each learner has a textbook, which is purchased by the learner. Ms. King has a range of texts and resources, including international texts, from which she and her colleagues develop and share worksheets6. Learners have access to a computer lab and so they are given projects to do, either using the mathematical software or using the internet as a research tool, for example a project on the history of mathematics. Mr. Daniels’ classroom has small tables and chairs, which are arranged in groups of four. The classroom is in good repair, and there are notice boards with a few math posters that Mr. Daniels has obtained. There is a teacher’s desk and one cupboard that overflows with supplies, worksheets, learners’ work, and other documents (there is no other storage space). Table 2.3  Description of research classes Teacher Grade Class size Tracking/level of class Mr. Nkomo 11 28 Untracked: mixed standard and higher gradea Mr. Mogale 11 43 Tracked: higher grade (top class) Mr. Daniels 11 35 Tracked: standard grade Mr. Peters 10 45 Untracked Ms. King 10 27 Tracked: second highest class of seven aAt the time of the study, the Grade 12 mathematics examination could be taken at two levels, standard and higher grade. Success on the higher grade (or exceptionally good marks on the stan- dard grade) granted access to scientific fields at university 6 Two mathematics teachers at this school developed a very strong curriculum of investigations in the 1980’s and Ms. King and her colleagues still use some of this (McLachlan & Ryan, 1994).

32 2  Contexts, Resources, and Reform There is a chalkboard and chalk and an overhead projector and screen, although electricity is not always available and so it does not always work. The school issues a textbook to each learner, although Mr. Daniels and his colleagues work predominantly from worksheets, which they develop and share. Mr. Nkomo and Mr. Mogale’s classrooms are similar. Tables are shared between two learners, with enough space for both. These are put together in pairs to form groups of four learners. There is a chalkboard and chalk, but no teacher’s desk and no overhead projector and screen. Mr. Nkomo’s classroom has a cup- board that stores cleaning supplies, but he has an office nearby where he keeps texts, worksheets, and learners’ work. Mr. Mogale keeps his work in the staff- room. Mr. Nkomo’s classroom is in good repair, although there is graffiti on the cupboard and notice boards. He occasionally puts posters and learners’ work on the notice boards but has to be careful because they often disappear. Electricity is regularly available. In Mr. Mogale’s classroom, some windows and the door are broken. There is no regular supply of electricity but on darker days (when it rains) lights can be provided with a starter. Both Mr. Nkomo’s and Mr. Mogale’s learn- ers are issued a textbook, although they work from worksheets most of the time. Mr. Peters’ classroom has an “old style” desk with adjoined chair for each learner, which means that they cannot easily be arranged in groups of more than two. There is a teacher’s desk and cupboard, a chalkboard and chalk, no overhead projector and screen and no electricity, so on rainy days the classroom is dark. Some windows are broken. The school has some textbooks but there are not enough for each learner and so they are not issued and Mr. Peters works from worksheets that he develops. I have spent some time describing the schools and the classrooms. This serves three purposes. First, it gives the readers a picture of the contexts so that they might better understand our later analyses of teaching and learning in these classrooms. Second, it shows that the five schools in which we worked are generally better resourced than most South African schools, although not as well resourced as many classrooms in “developed” countries. Third, it shows the differences in resources and socio-economic profiles across the five schools, which will allow me to make some claims in relation to equity and teaching and learning mathematical reason- ing. The five classrooms create a matrix design across grades and socio-economic status as described in Table 2.4. I will show later in the chapter how this design is both reinforced and complicated by learners’ knowledge in relation to their socio- economic status and by the tasks chosen by the teachers. Table 2.4  Variation across schools Socio-economic status Grade High Low Grade 11 Mr. Daniels Mr. Nkomo Mr. Mogale Grade 10 Ms. King Mr. Peters

Learner Knowledge 33 Learner Knowledge Learner knowledge was ascertained across the five classes through classroom observations and task-based interviews with learners. It was clear that Mr. Peters’ learners had extremely weak knowledge, probably a few grade levels below Grade 10. Mr. Nkomo’s learners were closer to grade level but showed some weaknesses, particularly in relation to Mr. Daniels’ learners, who were around grade level. This is somewhat surprising, given that Mr. Nkomo’s class was untracked and consisted of both higher and standard grade learners, while Mr. Daniels’ class was a standard grade class only (see Table 2.2). The learners’ knowledge in these cases reflects the socio-economic status of the schools. Further reflecting socio-economic status, Ms. Kings’ Grade 10 learners were extremely strong and were the second highest class in a strongly tracked grade. In fact, their knowledge was stronger than both Mr. Nkomo’s and Mr. Daniels’ Grade 11 learners. Mr. Mogale’s learners provide an interesting counterpoint to the SES/learner knowledge link. Although his is a low SES school, this class had been chosen in Grade 9 as the strongest learners in the grade, had been kept together as a class and had Mr. Mogale as their mathematics teacher since then. He had worked to build their mathematics knowledge and con- fidence over 3 years, informed by the principles of the new curriculum, which he had learned on in-service workshops. In the task-based interviews, two or three learners were interviewed together and encouraged to help each other. High-achieving learners were chosen from Mr. Nkomo, Mr. Peters, and Mr. Daniels’ classes while the learners from Mr. Mogale’s and Ms. King’s classes were close to average achievement. Mr. Peters’ learners showed very little facility in solving mathematics problems. Their solutions showed procedural errors in almost every step and suggested that they were looking for rules, rather than thinking about the meaning of what they were doing. Even the rules that they did remember, for example “what you do to the one side, you do to the other” were almost always applied incorrectly. Occasionally, with the easier problems, one learner used trial and error methods and obtained correct solutions but then did not know what to do with these, nor how to relate them to the mistaken rule-based calculations. Occasionally through prompting by the interviewer, this same learner was able to make some conceptual connections. The two learners did not manage to communicate with each other in ways that helped their problem solving; rather their conversations seemed to encourage even more mistakes and misunderstandings. Mr. Nkomo’s learners showed some facility with mathematical procedures and calculations and working algebraically without mistakes. However, they did not relate their calculations to the underlying mathematical meaning and when they were confronted with something slightly out of the ordinary, could not make sense of it. They did not use trial and error methods and were heavily dependent on the interviewer’s help to solve most of the problems. They were able to work together and sometimes help each other with procedural issues. The learners in Mr. Daniels’ class showed procedural fluency and were able to talk conceptually about the mathematical solutions they were developing. They were able

34 2  Contexts, Resources, and Reform to reason with mathematical objects, although some of their reasoning was flawed and somewhat problematic from a mathematical standpoint. They spent useful time talking and explaining ideas to each other, correcting mistakes and resolving conflict- ing ideas, while checking that their procedures were correct and eventually coming to consensus on most solution methods. They were able to work with the interviewer’s prompts and incorporate them into their own problem solving activities. The learners in Ms. King’s class were much more procedurally fluent with equa- tions than both Mr. Nkomo’s and Mr. Peters’ learners, and even slightly more fluent than Mr. Daniels’ learners even though they were a grade lower. They had been taught some Grade 11 concepts and procedures as “extras” in Grade 10 and were able to work with these as well, with some assistance from the interviewer. Conceptually, one learner struggled with some of the same issues that Mr. Daniels’ Grade 11 learners struggled with, while the other was able to reason mathematically in a particularly perceptive way. The two boys were able to work together and help each other. The learners in Mr. Mogale’s class were procedurally fluent. They made occa- sional mistakes, which they noticed themselves because they continuously checked their work, looking for mistakes. They also estimated answers as a check on their procedures. They understood the meanings of the mathematical objects they worked with and reasoned mathematically with them. They went further than the learners in the other classes, in that they noticed links with other areas of mathematics and posed interesting questions about their observations. So they extended their think- ing, creating new conjectures about the relationships between mathematical ideas. These differences in the learners’ knowledge were evident in the classroom interactions as well. These differences cannot be read as a comment on the particu- lar teachers in this study (except possibly in the case of Mr. Mogale who had taught these learners for 3 years). It is clear that both the strengths and the weaknesses in the learners’ knowledge comes from prior years of schooling and is a function of far more than only particular teachers. In four of the five classrooms, learners’ knowledge is strongly associated with the racial and socio-economic profiles of the schools. This makes sense because schools in poorer areas usually have fewer resources, larger classes, and generally, less knowledgeable teachers (Fleisch 2007). The strong association of learner knowledge with race and socio-economic status in this sample modifies the matrix design in Table 2.4 slightly (see Table 2.5 below). Given that Mr. Mogale’s learners provide a strong exception to the rule, being of low socio-economic status but with strong learner knowledge, we will be able to make some arguments which de-link learner knowledge and socio-economic status in subsequent chapters. Table 2.5  Variation across schools Strong/high Weak/low Learner knowledge/SES Grade Grade 11 Mr. Daniels Mr. Nkomo Mr. Mogale (knowledge) Mr. Mogale (SES) Grade 10 Ms. King Mr. Peters

The Tasks 35 The Tasks As part of the research design, the teachers worked together to plan tasks, which would engage learners in mathematical reasoning. The two Grade 10 teachers worked together and the three Grade 11 teachers worked together. They worked with drafts of new South African textbooks which were being developed in relation to the new curriculum (Jaffer and Johnson 2004; Johnson et al. 2006), as well as some of their own resources, in order to choose, modify, and develop tasks that they thought would be useful to elicit mathematical reasoning and would also fit in with their curriculum. They spent two sessions of 2½ h each, planning the tasks and how they might teach them. The Grade 11 Tasks The tasks developed by the Grade 11 teachers (see Appendix) aimed to get the learners to explore how horizontal and vertical shifts of a parabola on a Cartesian plane produce differences in the equations of the graphs. The task consisted of three activities. The first activity required the learners to trace a copy of the graph y = x2 onto a transparency, shift the transparency three units to the right and four units to the left, and observe what happened to the values of corresponding points on the shifted graphs in relation to the original graph. The second activity showed the original and the two shifted graphs on the plane, with their equations: y = (x − 3)2 and y = (x + 4)2 and asked learners to compare and contrast the graphs, and then to focus on the more general question of how the value of p in y = a(x − p)2 affects the graph. The third activity dealt with vertical shifts, with the graphs of y = (x − 3)2, y = (x − 3)2 + 2 and y = (x − 3)2 − 3. Again the learners were asked to compare and contrast the graph and then answer the more general questions of how the value of q in y = a(x − p)2 + q affects the graph. An analysis of the task using Stein et al.’s (2000) framework (see Chap. 3 for more detail) shows that that it demands higher level thinking from learners, pre- dominantly at level three – “procedures with connections”. According to Stein and her colleagues’ criteria, the activities suggest pathways, to follow, that are closely connected to underlying conceptual ideas, are represented in multiple ways to help learners build connections and develop meaning, and require learners to engage with conceptual ideas in order to complete the task successfully. Two additional aspects of the task are important for the subsequent analysis. First, the task is inductive, in that it asks learners to explore particular examples of shifting graphs and then make generalizations based on these examples. It does not ask for any form of deductive proof or justification. Exploratory questions such as “Discuss with a partner how these graphs differ and are the same” and “What do

36 2  Contexts, Resources, and Reform you observe?” are relatively open and unconstrained, except by the graphs, in what could count as an acceptable response. Learners could comment on only one obser- vation or on as many as they could find. They could comment with or without explicit justification. Second, the task contains a number of possibilities for misconceptions to arise and become visible. A key, counter-intuitive idea that is entailed in the task is that when the graph shifts in the positive direction, the equation has a negative sign in the brackets, i.e., y = (x − 3)2 is the equation of the graph that shifts three units to the right. Similarly, when the graph shifts in the negative direction, the sign in the brackets becomes positive. Many learners in all three Grade 11 classrooms demonstrated the misconception that the sign in the brackets should follow the direction of shift of the graph, making for some interesting discussions and exploration of the links between equations and graphs. Other conceptual issues that learners struggled with were: what does it mean if a graph extends infinitely along one axis; what counts as corresponding points; and how to read variables such as p and q in an equation. How these misconceptions influenced the teaching of mathematical reasoning in these classes is discussed in subsequent chapters. The Grade 10 Tasks Ms. King and Mr. Peters began their planning by looking for tasks that would enable learners to engage in all five strands of mathematical proficiency identified by Kilpatrick et al. (2001): conceptual understanding; procedural fluency; strategic competence; adaptive reasoning and productive disposition (see Chap. 6 for more detail). Ms. King wanted to focus on the integrated development of all five strands among her learners while Mr. Peters wanted to focus on the adaptive reasoning strand and develop his learners’ ability to justify their thinking. Given their slightly different foci, and because of Mr. Peters’ concerns that the tasks might be too chal- lenging for his learners, Ms. King and Mr. Peters used the same first task, but dif- ferent subsequent tasks. I will first discuss Ms. King’s tasks and then Mr. Peters’ tasks (see Appendix). Tasks 1 and 5 on Ms. King’s worksheet are primarily deductive tasks in that they require the learners to evaluate conjectures as true or false and then justify their decision. Task 1 asks whether x2 + 1 can equal zero and what the smallest value for x2 + 1 is if x is a real number. Task 5 asks whether n2 − n + 11 is a prime number, if n is a natural number. Learners might test the conjectures using specific examples. However, they do not need to, they could work on a general justification from the beginning. In the case of task 5, if they do test examples, the first 10 natural num- bers will give prime numbers but 11 will not and so makes the point that inductive testing is not good enough because there can be counter-examples. In this case, the general argument is that for n2 − n + p, n = p gives p2 which is not a prime number. Tasks 1 and 5 make demands on learners who are at level four (the highest level) of Stein et al.’s (2000) framework, which they call “doing mathematics”. The tasks

The Tasks 37 require nonalgorithmic thinking, they do not suggest specific solution approaches, they require learners to integrate existing knowledge to form understandings of new relationships, they require learners to examine task constraints, and they require some self-regulation and self-monitoring of the learners’ thinking processes. Tasks 2 and 3 require learners to work with the definition and meaning of a func- tion, and with function notation. They make level three demands in Stein and her colleagues’ hierarchy, suggesting solution methods that connect to underlying meanings and requiring multiple representations. Task 4 gives learners practice with function notation, which Ms. King thought was important in helping learners develop both conceptual understanding of and procedural fluency with functions. This task can be enacted at either level two or three of Stein’s hierarchy, depending on how individual learners approach it. The task can be approached using the pro- cedures of substitution and simplification of algebraic expressions without thinking much more deeply about the notion of a function. Alternatively, the task might sup- port learners to make connections with what they have done before, and come to a more fluent and better understanding of functions. Because making connections is not explicitly asked for in the task, this task would be considered to be a level two task – “procedures without connections”. Mr. Peters worked with the same first task as Ms. King, although he excluded the second part: what is the smallest value for x2 + 1. His first task read: Consider the following conjecture: “x2 + 1 can never be zero”. Prove whether this statement is true or false if x ∈ R. During the planning process, Mr. Peters expressed concern about how his learners would approach this task because of their weak knowledge. He worried about moving on with the same tasks as Ms. King, expecting that his learners would need more time working with the first task and that he would need to give them additional guidance. He developed a second task (Task 1B) where he scaffolded the learners’ substituting into various single-term expressions and working with the sign of the expression. After two lessons where learners struggled with Task 1, he decided that this task (1B) would not help, as learners tended to focus on the sign rather than the value of the expression. So while teaching and monitoring learners’ responses, he developed a second task that he hoped would address these difficulties, because the sign in front of each expression is not the sole determining factor of the sign of the expression (Task 2). Both Tasks 1 and 2 are primarily deductive and can be approached by using a combination of inductive and deductive methods. Mr. Peters hoped that both tasks would encourage the learners to use a combined inductive–deductive approach, through substituting, testing, and justifying conjectures. In Task 2, he had the more specific goals of developing procedural fluency in substituting into the expressions, conceptual understanding that the expressions represent a range of values, strategic competence in that learners should not read off whether the expression was positive or negative from superficial aspects of the expressions and adaptive reasoning in that they justified their answers. Task 1 would count as “doing mathematics” in Stein and her colleagues’ hierarchy, while Task 2, as Mr Peters intended it to be solved, would count as “procedures with connections”. In Task 2 there is a specified solution method (not in the task as such but Mr. Peters made it clear in class), which

38 2  Contexts, Resources, and Reform is intended to help learners make connections with underlying meanings and concepts. The above discussion of the tasks has shown differences in the ways they sup- port learners to make connections between procedures and meanings; integrate the various strands of mathematical proficiency; and how they constrain what might count as an acceptable solution. In subsequent chapters, we will show how the tasks afforded and constrained the contributions that learners made in the classroom and how the teachers dealt with these. For the purposes of this chapter, the discussion of tasks fills out the matrix design of the study in Tables 2.4 and 2.5. Given that the tasks used in each grade were the same or similar, comparisons within and across grades are made easier. The matrix design in Table 2.6 enables some extrication of the variables of task, learner knowledge, and socio-economic status in relation to the possibilities for teaching mathematical reasoning in differently resourced classrooms. All grade 11 teachers used the same tasks, which were inductive and which sup- ported procedures with connections to meaning. Mr. Daniels’ learners had strong mathematical knowledge and were of high SES while Mr. Nkomo’s learners had weak mathematical knowledge and were of low SES. Mr. Mogale’s learners pro- vide a contrast to the others in that they were of low SES but had strong mathematical knowledge. The two grade 10 teachers used similar tasks, which were mainly deductive and which varied from “doing mathematics”, through procedures with connections to procedures without connections. Ms. King’s learners were of high SES and had very strong mathematical knowledge while Mr. Peters’ learners were of low SES and had weak mathematical knowledge. These similarities and differ- ences among the teachers enable comparisons in relation to tasks, school context, and learner knowledge, which we pursue in Part 3 of this book. In Part 2, we look at the individual case studies of each of the teachers, which provide in-depth descriptions of their teaching of mathematical reasoning. Table 2.6  Variation across teachers in tasks, learner knowledge and SES Learner knowledge/SES Tasks Stronger/higher Weaker/lower Mr. Nkomo Grade 11 Mr. Daniels Mr Mogale (SES) Inductive Mr Mogale (knowledge) Procedures with connections Mr. Peters Grade 10 Ms. King Deductive (with some inductive) Procedures with and without connections, doing mathematics

Chapter 3 Mathematical Reasoning Through Tasks: Learners’ Responses In this chapter, I draw on the notion of mathematical reasoning discussed in Chap. 1 as well as the approach to mathematics in the new curriculum in South Africa. The new curriculum takes the view that mathematics should make sense to all learners and that learners should be given opportunities to solve problems, look for patterns, make conjectures, make inferences from data, explain and justify their ideas and challenge others’ ideas. The National Curriculum Statement Grades 10–12 (Department of Education 2003, pp. 9–10) describes the purpose of mathe- matics in the new curriculum as follows: Mathematics will enable learners to: • Communicate appropriately by using descriptions in words, graphs, symbols, tables, and diagrams • Use mathematical process skills to identify, pose, and solve problems creatively and critically • Organize, interpret, and manage authentic activities in substantial mathematical ways that demonstrate responsibility and sensitivity to personal and broader societal concerns • Work collaboratively in teams and groups to enhance mathematical understanding • Collect, analyse, and organize quantitative data to evaluate and critique conclu- sions arrived at • Engage responsibly in quantitative arguments relating to local, national, and global issues These broad curriculum outcomes resonate with the description of mathematical reasoning given in Chap. 1 and show the intention of the new curriculum to produce learners who can work with mathematics in a variety of ways. The intention has moved away from learners who use calculations and formulas as the only ways of solving mathematical problems, and producing only accepted correct solutions to problems. When our current learners leave school they will be facing a world dif- ferent from that of their parents and teachers. They will need the mathematical skills of reasoning and justification to respond to a range of challenges. We do not help our learners to rise to these challenges by teaching them that all problems can be solved merely through the application of certain procedures. The National Curriculum K. Brodie, Teaching Mathematical Reasoning in Secondary School Classrooms, 43 DOI 10.1007/978-0-387-09742-8_3, © Springer Science+Business Media, LLC 2010

44 3  Mathematical Reasoning Through Tasks: Learners’ Responses Statement argues that there are many ways that learners can learn mathematics successfully. Learners should be given opportunities to communicate their ideas critically and creatively as they work on mathematics tasks. One of the key influences in how learners learn to reason mathematically is the nature of the tasks that they work with in class (Stein et al. 1996, p. 72, 2000). In this chapter, I analyse Grade 11 learners’ work on tasks that involve mathematical reasoning. At the time of the study, the Grade 11 syllabus did not require learners to solve non-routine problems that they had not seen before and did not include tasks that required learners to display their reasoning, or to formulate, test, and justify conjectures. The tasks mostly required learners to carry out calculations or procedures that they had been taught. In my experience during the past 10 years of marking Grade 12 examination papers, I had come to realize that learners perform poorly on tasks that involve higher order mathematical reasoning. This chapter aims to explore the challenges that learners may encounter with tasks that involve mathematical reasoning and the ways in which a teacher can help learners to improve their mathematical reasoning. Tasks that Support Mathematical Reasoning Mathematical tasks are given to learners by the teacher to engage them in mathe- matical activity in order to develop certain mathematical concepts or practices. Stein et  al. (1996) defined a mathematical task as a classroom activity, which is intended to focus learners’ attention on a particular mathematical idea. Once the teacher has set up learning goals, s/he can give tasks that match with her/his goals for the kinds of thinking s/he would like the learners to engage in. If the teacher wants learners to memorize mathematical facts and procedures she/he will give tasks that require memorizing. The old curriculum tended to prioritize memorizing over other forms of mathematical activity and most textbook and examination tasks required learners to memorize and recall facts and procedures. The new curriculum requires a broader range of mathematical practices and if teachers are to help learners develop these, we will need to broaden the range of tasks that we ask learners to engage in. Stein et  al. (2000) distinguished between two levels of cognitive demand of mathematical tasks, and within each of these, two kinds of tasks. Lower level tasks are memorisation tasks and tasks that require procedures without connections to meaning or concepts. Higher-level tasks are those that require procedures with connections to meaning or concepts and “doing mathematics” tasks, which require a high level of exploration from learners. These are described below. Memorization tasks involve reproducing previously learned facts, rules, formulae, or definitions. Memorization tasks do not require any explanation from learners; they are straightforward and learners use well-known facts to solve them. Procedures without connection tasks require reproduction of procedures but without

Tasks that Support Mathematical Reasoning 45 connection to underlying concepts. Such tasks are focused on producing correct solutions rather than developing mathematical understanding. Procedures with connection tasks focus learners’ attention on the use of procedures for the purpose of developing deeper levels of understanding of mathematical concepts and ideas. Such tasks focus learners on the procedure of solving mathematics problems in a meaningful way. Doing mathematics tasks do not require any procedure to be followed. There is no predictable way of solving these problems. Learners working with such tasks need to analyse task constraints and creatively find their own solutions. Since learners interact with tasks, the cognitive demands of the tasks depend not only on the task, but also on the learner. For example, a task that is at a high level for a Grade 8 learner might be at a lower level for a Grade 11 learner; or two learn- ers in the same grade might solve the same task in different ways. One might use procedures and think about the connections of the procedures to the underlying mathematical concepts, while another might use the same procedures without making any connections to meaning or concepts. The cognitive demand of the tasks can be recognized in the task features, which include the “number of solution strategies, number and kind of representa- tions and communication requirements” (Stein et al. 1996, p. 455). Tasks that can be solved by using different approaches, and those that require learners to bring together different representations and to explain, justify, and communicate their ideas are likely to be of higher cognitive demand than those that have only one method of solution and do not require additional effort in working with represen- tations and explaining ideas. In order to encourage mathematical reasoning as suggested by the new curriculum, teachers should give learners tasks that allow for different levels of engagement, including the higher levels. If learners are given only tasks that are of a lower level, they will find it difficult to tackle higher-level tasks. Examples of lower-level tasks are found in many textbooks, for example “Sketch the graph of y = (x − 3)2 + 2”. In response to such tasks most learners will use the usual procedure of finding the x and y intercepts, the turning point and draw the graph. Most learners use these procedures without understanding the relation- ship between the equation and the graph, however as mentioned above, some (very few) learners do make connections with the meaning of the graph. An example of a higher-level task from a new curriculum textbook (Bennie 2006), explicitly asks learners to make connections between procedures and meanings and to justify conjectures: (a) Draw a sketch of y = x2. Use a table if necessary. (b) Consider the graphs with equations y = ax2. Make a conjecture about the effect on the graph when you change the value of a. Test your conjecture by drawing the graphs of y = 2x2, y = 3x2, and y = 1/2x2 on the same system of axes you used in (a). (c) What will happen if the value of a in y = ax2 is negative? Choose suitable values for a and test your conjecture. (d) Summarize your observations in questions a and b above, etc.

46 3  Mathematical Reasoning Through Tasks: Learners’ Responses This task requires learners to go beyond drawing the graph and to make conjectures from the graph. It also requires them to summarize their observations, which is something that was not visible in the old curriculum. Stein et  al. (1996, 2000) distinguish between two phases of task-use in the classroom: task set-up and task implementation. They note that the task demands can shift between the set-up and implementation phases, depending on how learners engage with the task and how teachers interact with learners. Often tasks that are set up at a higher level, decrease in level as they are implemented (Modau and Brodie 2008; Stein et al. 1996, 2000). This can happen for a number of reasons: learners might choose to work at a lower level than the task requires, ignoring the higher level task demands; and teachers might give learners too much help, which reduces the level, for example “funnelling” the task (Bauersfeld 1988). Stein et al. (1996) argue that “classroom norms, task condi- tions and teachers’ and students habits and dispositions” (p. 461) can all influ- ence how tasks change at implementation. Their research shows that tasks are usually maintained at the same level or decline in level. The higher-level tasks in particular tended to decline. Teaching for Mathematical Reasoning Stein et al.’s task framework is consistent with both constructivist and socio-cultural perspectives on learning, which were discussed in Chap. 1. The notion of cognitive demand suggests individual or group engagement with tasks that promote learning, thinking, and reasoning at different levels in the individual. The idea that the level of the task changes in interaction between teacher and learner, or among learners suggests a socio-cultural perspective, because social aspects of the classroom determine the level of individual cognition. These theoretical frameworks suggest important implications for teachers. Constructivism shows how errors and misconceptions can be deep-seated in learners’ conceptual structures and learners have to do the difficult work of trans- forming their own thinking, with their teacher’s help. If learners’ responses are incorrect, it does not necessarily help to tell them so. As Heaton (2000) remarks: “telling him he was wrong would not necessarily change how he thought”. Heaton also goes on to say that if learning and teaching are about understanding why answers are right or wrong, then it is important to explore learners’ errors with them. The teacher can ask learners to explain their answers by posing ques- tions like: “can you explain how you got the answer” or “convince me that your answer is correct”. Moreover, asking learners to explain is just as important when their answers are correct as when they are incorrect because explaining can deepen their own thinking, and help others in the class. It also encourages the “social norm” (Yackel and Cobb 1996) that all answers should be justified in mathematics.

The Classroom and the Tasks 47 A socio-cultural perspective suggests that by asking for explanation and justification, the teacher can make learners aware of their reasoning and support them to con- struct appropriate mathematical ideas. The teacher can, through questions and prompts, try to provoke learners into thinking in particular ways and support them to compare, verify, explain, and justify their conjectures. It is not easy for the teacher to ask questions that will make learners aware of their reasoning, since learners might not respond as the teacher expects. Ball (2003) argues that when a teacher whose students have never been asked to explain their thinking asks them to justify their solution, s/he is likely to be greeted with silence. When s/he asks a learner to explain her/his method, the learner will probably think that she/he made an error. It is therefore important, but demanding, for teachers to develop norms of interaction in their classrooms (Yackel and Cobb 1996, see Chap. 1 for more discussion). So teaching for mathematical reasoning involves teachers being aware of the type of tasks that we give to learners, choosing tasks that enable learners to make sense of mathematics and that give them opportunities to investigate, analyse, explain, conjecture and justify their thinking, and interacting with learners around the tasks to maintain, or even raise, the level of the task. The Classroom and the Tasks For the purposes of linking this chapter with chapter two, it is important to note that my pseudonym in the study is Mr Nkomo. This study was conducted in one of my Grade 11 classes, in a functional township school west of Johannesburg, with very basic facilities (see Chap. 2 for more detail). All the teachers and learners in the school were “black African” South Africans (see Chap. 2). English is not the main language of any of us, but all teaching and learning of mathematics occurs in English. There were 1,650 learners in the school at the time of the study and 42 teachers, giving a teacher–learner ratio of 1:39. There were 28 learners in my Grade 11 class. Since fewer learners take mathematics in the higher grades, these classes tend to be smaller. The learners worked in ten groups of between 2 and 4 learners in a group. These groups were established prior to the study and learners were used to working with each other. The class was of mixed ability in mathematics and learners in the class were taking mathematics on both higher grade/standard grade.1 I planned a series of tasks on functions with the two other Grade 11 teachers as part of this project (see Chap. 2 and Appendix). The tasks were planned to take 1 week of class time and were intended to engage learners’ mathematical reasoning as 1 As explained in Chap. 2, at the time of the study, the Grade 12 mathematics examination could be taken at two levels, standard and higher grade. Success on the higher grade (or exceptionally good marks on the standard grade) granted access to scientific fields at university.

48 3  Mathematical Reasoning Through Tasks: Learners’ Responses discussed below. As learners worked in their groups, I went around asking questions where necessary. I also conducted whole class discussions after each task. At the end of each lesson, I took in each group’s work and read it carefully, preparing how to conduct a whole class discussion on different groups’ work the next day. In the next section, I will analyse the first task and so I describe it here (see Appendix for actual task). Learners were presented with the graph of y = x2, asked to move it first 3 units to the right and then 4 units to the left. In each case, they were asked to compare the turning points of the two graphs (y = x2 and each shifted graph), and after that to compare corresponding points on the two graphs. The task encouraged learners to experiment with shifting graphs, and to focus on particular points in order to begin to notice relationships between them. Since they were explicitly told how to do this, i.e. told first to look at the turning point and then at other points, and since they were given a table with values for y = x2 filled in, which they had to complete, this is a “procedures with connections” task in Stein et al.’s (2000) framework. Learners were asked to write down their observations about how the points shifted as the graphs shifted and in doing so to make connections between the shifting graphs and shifting values of co-ordinates. Learners had to examine parameters, experiment with shifting graphs, make conjectures and write down observations. Learners’ Responses: An Overview My first step was to analyse the learners’ written responses to the tasks in three categories: 1. Comparing the turning point of y = x2 to the turning point of the new graphs 2. Choosing and comparing other corresponding points on the graph 3. Observations about the corresponding points of the two graphs Table 3.1 shows the total number of groups who gave correct and incorrect responses for each of the above (there were ten groups). Table 3.1 shows that learners were able to identify and compare the new turning points of the graphs, but struggled to identify, compare and make observations about other corresponding points on the graphs. However, as discussed above, correct and incorrect responses are not sufficient to illuminate or develop learners’ mathematical reasoning. So I continued to analyse learners’ reasoning, both from the written tasks that they handed in, and from videotapes of whole class discussions. Table 3.1  Correct and incorrect responses Category 1 Category 2 Category 3 Correct 10 1 0 Incorrect 0 9 10